\draw[-, ultra thick] (-2.5,2.5) -- (-1.5,2.5) -- (-1,3) -- (0,2) -- (.5,2.5) -- (2.5,2.5);
\draw[-, dashed] (-2.5,1.5) -- (-.5,1.5) -- (0,2) -- (2.5, -0.5);
\draw[-] (-2.5,0.5) -- (.5,0.5) -- (1.5,1.5) -- (2.5, 1.5);
\draw[-] (-2.5,-0.5) -- (.5,-0.5) -- (1.5, -1.5) -- (2.5,-1.5);
\draw[-, ultra thick, dashed] (-2.5,-1.5) -- (-.5,-1.5) -- (0,-2) -- (2.5,.5);
\draw[-, ultra thick] (-2.5,-2.5) -- (-1.5,-2.5) -- (-1,-3) -- (0,-2) -- (.5,-2.5) -- (2.5,-2.5);
\draw[-, ultra thick, dotted] (-2.5,-3.5) -- (-2,-4) -- (-1,-3) -- (-.5,-3.5) -- (2.5,-3.5);
\draw[-] (-2.5,-4.5) -- (-2,-4) -- (-1.5,-4.5) -- (2.5,-4.5);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale=.5,baseline=0]%s2s1s0
\node at (-3.1,4.5) {$\tau_5$};
\node at (-3.1,3.5) {$\tau_4$};
\node at (-3.1,2.5) {$\tau_3$};
\node at (-3.1,1.5) {$\tau_2$};
\node at (-3.1,0.5) {$\tau_1$};
\node at (-3.1,-0.5) {$\tau_{\ol1}$};
\node at (-3.1,-1.5) {$\tau_{\ol2}$};
\node at (-3.1,-2.5) {$\tau_{\ol3}$};
\node at (-3.1,-3.5) {$\tau_{\ol4}$};
\node at (-3.1,-4.5) {$\tau_{\ol5}$};
\draw[-] (-2.5,4.5) -- (-2,4) -- (-1.5,4.5) -- (2.5,4.5);
\draw[-, ultra thick, dotted] (-2.5,3.5) -- (-2,4) -- (-1,3) -- (-.5,3.5) -- (2.5,3.5);
\draw[-, ultra thick] (-2.5,2.5) -- (-1.5,2.5) -- (-1,3) -- (0,2) -- (.5,2.5) -- (2.5,2.5);
\draw[-, dashed] (-2.5,1.5) -- (-.5,1.5) -- (0,2) -- (2,0) -- (2.5,.5);
\draw[-] (-2.5,0.5) -- (.5,0.5) -- (1.5,1.5) -- (2.5, 1.5);
\draw[-] (-2.5,-0.5) -- (.5,-0.5) -- (1.5, -1.5) -- (2.5,-1.5);
\draw[-, ultra thick, dashed] (-2.5,-1.5) -- (-.5,-1.5) -- (0,-2) -- (2,0) -- (2.5,-.5);
\draw[-, ultra thick] (-2.5,-2.5) -- (-1.5,-2.5) -- (-1,-3) -- (0,-2) -- (.5,-2.5) -- (2.5,-2.5);
\draw[-, ultra thick, dotted] (-2.5,-3.5) -- (-2,-4) -- (-1,-3) -- (-.5,-3.5) -- (2.5,-3.5);
\draw[-] (-2.5,-4.5) -- (-2,-4) -- (-1.5,-4.5) -- (2.5,-4.5);
\end{tikzpicture}\,.
\end{equation}
Two $\pi$-bitableaux and two $\tau$-bitableaux of shape $(21,11)$
\begin{equation}\label{eq:2111fex}
\left(\, \tableau[scY]{\pi_5 | \pi_2,\pi_1}\, {,}\,
\tableau[scY]{\pi_4 | \pi_3}\, \right)\ntnsp {,}
\qquad
\left(\, \tableau[scY]{\pi_5 | \pi_2,\pi_3}\, {,}\,
\tableau[scY]{\pi_1| \pi_4}\, \right)\ntnsp {,}
\qquad
\left(\, \tableau[scY]{\tau_5 | \tau_1,\tau_2}\, {,}\,
\tableau[scY]{\tau_4 | \tau_3}\, \right)\ntnsp {,}
\qquad
\left(\, \tableau[scY]{\tau_4 | \tau_1,\tau_2}\, {,}\,
\tableau[scY]{\tau_5 | \tau_3}\, \right)\ntnsp {.}
\end{equation}
%  \left(\, \tableau[scY]{\pi_5 | \pi_2,\pi_1}\,\raisebox{-4mm}{,}\,
%       \tableau[scY]{\pi_4 | \pi_3}\, \right)\ntnsp\raisebox{-4mm}{,}
%  \qquad
%  \left(\, \tableau[scY]{\pi_5 | \pi_2,\pi_3}\,\raisebox{-4mm}{,}\,
%  \tableau[scY]{\pi_1| \pi_4}\, \right)\ntnsp\raisebox{-4mm}{,}
%  \qquad
%  \left(\, \tableau[scY]{\tau_5 | \tau_1,\tau_2}\,\raisebox{-4mm}{,}\,
%  \tableau[scY]{\tau_4 | \tau_3}\, \right)\ntnsp\raisebox{-4mm}{,}
%  \qquad
%  \left(\, \tableau[scY]{\tau_4 | \tau_1,\tau_2}\,\raisebox{-4mm}{,}\,
%  \tableau[scY]{\tau_5 | \tau_3}\, \right)\ntnsp\raisebox{-4mm}{.}
Since paths $\pi_2$ and $\tau_2$
%are grounded (
intersect $\pi_{\ol2}$ and $\tau_{\ol2}$, respectively, they are grounded
and must appear in the left tableaux.
%($\star$ raise boxes on the commas below too?)
%. Change path families to type $2134$ and $2\ol134$
%so that tableaux can be cylindrical?)
The poset $Q(2345\ol1)$ and four marked $Q(2345\ol1)$-bitableaux of shape
$(21,11)$ are
%and a few marked
\begin{equation}\label{eq:2111pex}
\begin{tikzpicture}[scale=.5,baseline=10]%s2s22
\draw[fill] (0,3) circle [radius=0.15];
\draw[fill] (0,1) circle [radius=0.15];
\draw[fill] (0,-1) circle [radius=0.15];
\draw[fill] (1,2) circle [radius=0.15];
\draw[fill] (1,0) circle [radius=0.15];
%  \draw[thick] (1,0) circle [radius=0.35];
\draw[thick] (0,-1) circle [radius=0.35];
\node at (-.5,3) {$\scriptstyle 5$};
\node at (-.5,1) {$\scriptstyle 3$};
\node at (-.6,-1) {$\scriptstyle 1$};
\node at (1.5,2) {$\scriptstyle 4$};
\node at (1.5,0) {$\scriptstyle 2$};
\draw[-, thick] (0,3) -- (0,-1);
\draw[-, thick] (0,3) -- (1,0);
\draw[-, thick] (1,2) -- (0,-1);
\draw[-, thick] (1,2) -- (1,0);
%\node at (0.5,-4.5) {$Q(\ol13\ol2)$};
\end{tikzpicture}\ntksp ,
\qquad
\left(\, \tableau[scY]{3 | 1^\star,2}\, {,}\, \tableau[scY]{5 | 4}\, \right)\ntnsp {,}
\quad
\left(\, \tableau[scY]{3 | 1,4}\, {,}\, \tableau[scY]{5 | 2}\, \right)\ntnsp {,}
\quad
\left(\, \tableau[scY]{3 |2,1}\, {,}\, \tableau[scY]{4 | 5}\, \right)\ntnsp {,}
\quad
\left(\, \tableau[scY]{1^\star | 3,5}\, {,}\, \tableau[scY]{4 | 2}\, \right)\ntnsp {.}\nTksp\nTksp
%  \left(\, \tableau[scY]{3 | 1^\star,2}\,\raisebox{-4mm}{,}\, \tableau[scY]{5 | 4}\, \right)\ntnsp\raisebox{-4mm}{,}
%  \quad
%  \left(\, \tableau[scY]{3 | 1,4}\,\raisebox{-4mm}{,}\, \tableau[scY]{5 | 2}\, \right)\ntnsp\raisebox{-4mm}{,}
%  \quad
%  \left(\, \tableau[scY]{3 |2,1}\,\raisebox{-4mm}{,}\, \tableau[scY]{4 | 5}\, \right)\ntnsp\raisebox{-4mm}{,}
%  \quad
%  \left(\, \tableau[scY]{1^\star | 3,5}\,\raisebox{-4mm}{,}\, \tableau[scY]{4 | 2}\, \right)\ntnsp\raisebox{-4mm}{.}\nTksp\nTksp
\end{equation}
%($\star$ Use ytableau package to make tableaux line up better?)
%  \bigg(
%  {\sum_\sum} \bigg),

If $(U,V)$ is an $F_w$-bitableau or $Q(w)$-bitableau,
then $U$, $V$ could have some of the properties enumerated
between (\ref{eq:posettableaux}) and
Some combinations of these
%($\star$ Give examples of marked $F$-bitableaux
%of different shape and having
%some of the properties below, and a couple of entries $T_{i,j}$.)
%corresponding to earlier
%used marked $F$-bitableaux?)
%in which marked elements of $Z$ may appear only in $U$,
%and may be circled there.  {\em Types} of path families in $U$ and $V$
%are given by elements of $\mfs a$ and $\mfs{n-a}$, respectively.
%$\star$ Define a bitableau so that only paths indexed by $1,\dotsc, n$
%appear.  If some path $\pi_i$ ($i > 0$) has sink $\ol j$ ($j > 0$), then
%let $v$ be the vertex where $\pi_i$ and $\pi_{\ol i}$ intersect,
%and define $\ol{\pi_i}$ to be the path which consists of the source-$i$-to-$v$
%portion of $\pi_i$, concatenated to the $v$-to-sink-$j$ portion of $\pi_{\ol i}$.
%Call this a ``marked path''.
%Several properties which $\pi$-tableaux ($Q$-tableaux)
%may possess
%can be
%defined for $P$-tableaux where $P$ is any poset.
%and their properties
%are enumerated between (\ref{eq:posettableaux}) and
may be used to interpret trace evaluations as in
Theorems~\ref{t:epsiloneta} -- \ref{t:psi}.
%Define a $P$-descent in $T$ to be a pair $(i,j)$ such that
%$T_{i,j} >_{P} T_{i,j+1}$ and let $\des_{P}(T)$ be
%the number of $P$-descents in $T$.  Define $\ol{T}_{i,j}$ to be the
%entry of $T_i$ whose label is $j$th smallest (as an integer).
%Define a $P$-excedance in $T$ to be a pair $(i,j)$ such that
%$T_{i,j} >_P \ol T_{i,j}$ and let $\exc_P(T)$ be the number of
%$P$-excedances in $T$.
%$\star$ Define the poset $Q(\pi)$ (and other concepts below) somewhere.
%$\star$ Define {\em marked $Z$-bitableaux}, {\em marked $P$-bitableaux},
%Given a marked $P$-bitableau $(U,V)$ and $T \in \{ U,V \}$,
%call the tableau $T$
%\item {\em column strict} if the entries of each column satisfy
%  $T_{i,j} <_P T_{i+1,j}$,
%\item {\em descent-free} or {\em row-semistrict}
%  if $\des_{Q(\pi)}(T) = 0$,
%\item {\em standard} if it is column-strict and row-semistrict,
%\item {\em row-closed} if in each row, the set of
%  source indices and the set of sink indices are equal, up to absolute value.
%\item {\em left row-strict} if the path indices in each row increase from left
%  to right,
%\item {\em excedance-free} if $\exc_P(T) = 0$.
%($\star$ Give examples of different types of tableaux.)
%(Or figure out how to ``box'' a path in $Z(w)$.)

\begin{thm}\label{t:epsiloneta}
Let $w \in \bn$ \avoidp{} and let $Q = Q(w)$ be the
related type-$\msfBC$ unit interval order.
%  be $\msfBC$-smooth with zig-zag network $Z = Z(w)$, and
For each bipartition $(\lambda,\mu) \vdash n$ we have
%  and let CS and RSS denote the column-strict and row-semistrict properties,
%  respectively.
%abbreviate the properties
%  \text{ CS } = \text{ column-strict of type $e$},
%  \text{ RSS } = \text{ row-semistrict of type $e$},
%  \end{equation*}
%    define the set $\mathcal S$ of three tableau properties
%  \begin{equation*}
%    \mathcal S = \{ \text{ row-semistrict of type $e$},
%                    \text{excedance-free of type $e$},
%                    \text{row-closed and left-row-strict } \}.
%  \end{equation*}
%  We have
%  We may express the evaluations at $\btc w1$ of
%  induced one-dimensional characters in terms of
%  the zig-zag network $Z = Z(w)$, column-strictness, and row-semistrictness
%  as follows.
\begin{equation*}
\begin{aligned}
%  \begin{enumerate}[1.]
%  \item
%  \noindent
(\epsilon\epsilon)^{\lambda,\mu}(\btc w1) &=
%    \begin{equation*}
\# \{ (U,V) \in \bitab(Q,\lambda^\tr,\mu^\tr) \,|\,
U, V\, \text{column-strict\,} \},\\
%    \end{equation*}
%    \item
%For any property $i \in \mathcal S$,
%      \noindent
(\epsilon\eta)^{\lambda,\mu}(\btc w1) &=
\#\{(U,V) \in \bitab(Q,\lambda^\tr,\mu) \,|\,
U\, \text{column-strict}, V\, \text{row-semistrict\,} \},\\
%      \noindent
%      \item %For any property $i \in \mathcal S$,
%    \begin{equation*}
(\eta\epsilon)^{\lambda,\mu}(\btc w1) &=
\# \{ (U,V) \in \bitab(Q,\lambda,\mu^\tr) \,|\,
U\, \text{row-semistrict}, V\, \text{column-strict\,} \},\\
%    \end{equation*}
%  \item %For any properties $i, j \in \mathcal S$,
%    \begin{equation*}
(\eta\eta)^{\lambda,\mu}(\btc w1) &= \# \{ (U,V) \in \bitab(Q,\lambda,\mu) \,|\, U, V\, \text{row-semistrict\,} \}.
\end{aligned}
\end{equation*}
%  \end{enumerate}
\end{thm}
\begin{proof}
%Let $Z= Z(w)$ be the network corresponding to $w \in \bn$,
Let $F = F_w$ have path matrix $A$,
fix $(\lambda,\mu)\vdash n$ with $|\lambda| = k$,
and let $\theta = (\zeta \xi)^{\lambda,\mu}$ be one of the characters
in the theorem.
%be the path matrix of $F = F_w$,
%  fix $\zeta, \xi \in \{ \epsilon, \eta \}$, and define
%  $\theta = (\zeta \otimes \delta \xi) \upparrow_{\mfb{\lambda,\mu}}^{\bn}$,
%  with $|\lambda| = k$.
By Lemma~\ref{l:tensorimm} and Theorem~\ref{t:charevalimmbc}, we have
\begin{equation}\label{eq:immchi}
(\zeta\xi)^{\lambda,\mu}(\btc w1) =
%    \bnimm{\theta}(A) =
\bnimm{(\zeta\xi)^{\lambda,\mu}}(A) =
%\imm{\chi}(A) =
\sumsb{I \subseteq [n]\\|I|=k} \simm{k}{\zeta^\lambda}(A^+_{I,I})
\simm{n-k}{\xi^\mu}(A^-_{[n] \ssm I,[n] \ssm I}).
\end{equation}
%  where the sum is over all
%  $|\lambda|$-element subsets $I$ of $[n]$.
%  $k$-element subsets $I$ of $[n]$.
By (\ref{eq:lmw}), each of the type-$\msfA$ immanants above is a
sum of products of determinants
%, if $\zeta$ or $\xi$ is $\epsilon$,
or permanents
%, if $\theta_i = \eta$)
of matrices in
$\{ A^+_{J,J} \,|\, J \subset I \}$
or $\{ A^-_{J,J} \,|\, J \subset [n] \ssm I \}$.
By Proposition~\ref{p:colstrictetc} each
%of these factors
factor counts
certain one-row or one-column tableaux and their product counts
bitableaux of the required shape.
%($\star$ Mention bitableaux?)
%and (\ref{eq:lmw}),
%the results follow.
%, we have that
%$\imm{\epsilon^\lambda}(A^+_{I,I})$ equals the number of
%column-strict $Q_I$-tableaux of shape $\lambda^\tr$
%containing at most one grounded element per column, and such an element
%may be marked.  On the other hand,
%$\imm{\epsilon^\lambda}(A^-_{[n]\ssm I,[n]\ssm I})$ equals the number of
%column-strict $Q_{[n]\ssm I}$-tableaux
%of shape $\mu^\tr$
%containing no grounded elements.
%By Corollary~\ref{c:colstrictetc},
%$\imm{\eta^\lambda}(A^+_{I,I})$ equals the number of
%descent-free $Q_I$-tableaux of shape $\lambda$
%in which any number of grounded paths may be marked.
%On the other hand, $\imm{\eta^\lambda}(A^-_{I,I})$ equals the number of
%$Q_I$-tableaux of shape $\lambda$ containing no grounded elements.
%($\star$ Can this proof be shortened?)
%($\star$ Is it necessary to say more?)
\end{proof}

\begin{cor}\label{c:epsiloneta}
Let $w \in \bn$ \avoidp{} and have zig-zag network $F_w$ and
type-$\msfBC$ unit interval order $Q = Q(w)$.
In Theorem~\ref{t:epsiloneta}
%  column-strict $F_w$-tableaux of type $e$ with column-strict $P$-tableaux.
we may replace
(marked) column-strict $Q$-tableaux of shape $\nu$ with
(marked) colorings of $\inc(Q)$ of type $\nu$.
%  ($\star$ Make this precise.)
We may also replace
(marked) row-semistrict $Q$-tableaux of shape $\nu$
with either of the following:
\begin{enumerate}[(i)]
%    \item %property of $P(w)$ tableaux
%  may be replaced with the
%      excedance-free $F_w$-tableaux of type $e$,
\item (marked) excedance-free $Q$-tableaux of shape $\nu$,
\item (marked) row-closed, left row-strict $F_w$-tableaux of shape $\nu$.
\end{enumerate}
We also have that $(\eta\eta)^{\lambda,\mu}(\btc w1)$ equals
\begin{enumerate}[(i)]
%  \ (iii)
\item[(iii)] the number of
marked acyclic orientations of subgraph sequences
\end{enumerate}
\begin{equation}\label{eq:subgraphseq}
(\inc(Q_{I_1}), \dotsc, \inc(Q_{I_r}), \inc(Q_{J_1}), \dotsc, \inc(Q_{J_t}))
\end{equation}
%  of subgraphs of $\inc(Q)$
$(I_1,\dotsc,I_{\ell(\lambda)}, J_1,\dotsc,J_{\ell(\mu)})$
varies over all ordered set partitions of $[n]$ of type
\begin{equation*}
\end{equation*}
%$[|\lambda|]$ of type $\lambda$,
%$(J_1,\dotsc,J_{\ell(\mu)})$ is an ordered set partition of
%$[|\lambda| + 1, n]$ of type $\mu$,
%sequences of subgraphs
and all grounded vertices appear in $I_1,\dotsc,I_{\ell(\lambda)}$.
%  $\inc(Q)$, as described in
%  ($\star$ See \cite[\S 3.3]{SkanCCS}?)
%  \end{enumerate}
\end{cor}
\begin{proof}
% ($\star$ Iron out details of this proof.)
%This follows from
By Proposition~\ref{p:colstrictetc},
one-rowed tableaux
in Theorem~\ref{t:epsiloneta}
which are row-semistrict
correspond bijectively to one-rowed tableaux
with properties
$(i)$ or $(ii)$
%  to those
and to acyclic orientations $(iii)$
%  of the corresponding subgraphs
of $\inc(Q)$.
%  Also by Proposition~\ref{p:colstrictetc} these correspond to the
%  claimed multiples of left-anchored, row-semistrict tableaux. ($\star$ Say
%  this better.)
Thus there is a correspondence of
several-rowed tableaux and of acyclic orientations of subgraph sequences.
%correspond as well
In all cases, we have equality of
the indices of paths, poset elements, and vertices which are grounded,
so markings correspond in the obvious way.
%  by Theorem~\ref{t:wtc1interpssubg},  correspond bijectively to these as well.
\end{proof}
For example, consider evaluating traces at $\btc{2345\ol1}1$ by
applying Theorem~\ref{t:epsiloneta}
to the poset $Q(2345\ol1)$ in (\ref{eq:2111pex}).
The first bitableau in (\ref{eq:2111pex})
contributes to
$(\epsilon\eta)^{21,11}(\btc{2345\ol1}1)$ and
the second to $(\epsilon\epsilon)^{21,2}(\btc{2345\ol1}1)$,
the third to $(\eta\eta)^{21,11}(\btc{2345\ol1}1)$,
and the fourth to $(\eta\epsilon)^{21,2}(\btc{2345\ol1}1)$
and $(\eta\eta)^{21,11}(\btc{2345\ol1}1)$.
%The same is true
If we modify the criteria of the theorem
as in Corollary~\ref{c:epsiloneta} (i),
the same is true,
since a tableau with two or
fewer columns is $Q$-excedance-free if and only
if it is $Q$-row-semistrict.

If instead we modify the criteria of the theorem
as in Corollary~\ref{c:epsiloneta} (ii) we may
apply these to path families
covering $F_{2345\ol1}$ in (\ref{eq:pathex}).
In this case, we find that the third and fourth
tableaux in (\ref{eq:2111fex})
contribute to $(\eta\eta)^{21,11}(\btc{2345\ol1}1)$,
and that the fourth tableau also contributes to
The tableaux contribute to no additional evaluations listed in
Theorem~\ref{t:epsiloneta} because
the left tableau of the first bitableau is not left row-semistrict,
the left tableau of the second bitableau is not row-closed,
and the left tableaux of all four bitableaux fail to have the property
%    contributes to
%    $(\epsilon\eta)^{21,11}(\btc{2345\ol1}1)$ or
%    $(\epsilon\epsilon)^{21,2}(\btc{2345\ol1}1)$ because paths in the
%    left tableaux do not have the property
that source and sink indices are equal.
%Also observe that

Finally, we may modify the criteria of the theorem
by applying Corollary~\ref{c:epsiloneta} (iii) to
%    $\inc(Q(2345\ol1))$
\begin{equation}\label{eq:incQ23451}
\inc(Q(2345\ol1)) =
\begin{tikzpicture}[scale=.5,baseline=-3]%s2s22
\draw[thick] (0,0) circle [radius=0.35];
\draw[fill] (0,0) circle [radius=0.15];
\draw[fill] (1.5,0) circle [radius=0.15];
\draw[fill] (3,0) circle [radius=0.15];
\draw[fill] (4.5,0) circle [radius=0.15];
\draw[fill] (6,0) circle [radius=0.15];
\node at (0,.65) {$\scriptstyle 1$};
\node at (1.5,.65) {$\scriptstyle 2$};
\node at (3,.65) {$\scriptstyle 3$};
\node at (4.5,.65) {$\scriptstyle 4$};
\node at (6,.65) {$\scriptstyle 5$};
\draw[-] (0,0) -- (6,0);
%\draw[-, thick] (0,3) -- (1,0);
%\draw[-, thick] (1,2) -- (0,-1);
%\draw[-, thick] (1,2) -- (1,0);
%\node at (0.5,-4.5) {$Q(\ol13\ol2)$};
\end{tikzpicture},
\end{equation}
%= 1 -- 2 -- 3 -- 4 -- 5$,
and by marking and acyclically orienting sequences of subgraphs on
$(2, 1, 1, 1)$ vertices.  Two sequencces contributing to
$(\eta\eta)^{21,11}(\wtc{2345\ol1}1)$ are
%We have
%Sequences such as
\begin{equation}\label{eq:markedor}
\qquad
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[thick] (1,0) circle [radius=0.35];
\draw[fill] (1,0) circle [radius=0.15];
\draw[fill] (2.5,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 1$};
\node at (2.5,.65) {$\scriptstyle 2$};
\draw[->] (1,0) -- (2.3,0);
\end{tikzpicture}\ntnsp,\,\;
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[fill] (1,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 4$};
\end{tikzpicture}\ntnsp,\,\;
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[fill] (1,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 5$};
\end{tikzpicture}\ntnsp,\,\;
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[fill] (1,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 3$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[fill] (1,0) circle [radius=0.15];
\draw[fill] (2.5,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 3$};
\node at (2.5,.65) {$\scriptstyle 2$};
\draw[->] (1,0) -- (2.3,0);
\end{tikzpicture}\ntnsp,\,
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[thick] (1,0) circle [radius=0.35];
\draw[fill] (1,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle ^{\phantom{\star}}1^\star$};
\end{tikzpicture}\ntksp,\;
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[fill] (1,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 4$};
\end{tikzpicture}\ntnsp,\,\;
\begin{tikzpicture}[scale=.5,baseline=-3]
\draw[fill] (1,0) circle [radius=0.15];
\node at (1,.65) {$\scriptstyle 5$};
\end{tikzpicture}
\end{equation}
%    with both sequences contributing to

%    ($\star$ Finish with next criterion in the corollary.)
%($\star$ Think of some bitableaux that contribute to one trace evaluation
%but not another.  Do we need a bigger example?)  ($\star$ Which trace evaluations
%need examples, one from each theorem?)  ($\star$ Leave some evaluations
%as an exercise for the reader.)

%For example, we have $(\eta\epsilon)^{2,2}(\btc{234\ol1}{1}) = 2$,
%because only the first two bitableaux in (\ref{eq:112examples})
%are row-semistrict (also excedance-free) on the left, and column-strict
%on the right.

%These contribute to trace evaluations as follows:
%\item $(\epsilon\epsilon){11,2}(\btc{234\ol1}{1})$:
%\item $(\epsilon\eta){11,11}(\btc{234\ol1}{1})$:
%\item $(\eta\epsilon){2,2}(\btc{234\ol1}{1})$:
%\item $(\eta\eta){2,11}(\btc{234\ol1}{1})$:

%($\star$ Make sure that $Q$ is defined for all \pavoiding elements of $\bn$.
%Make sure that $F_w$ is defined for these elements as well.)

\begin{thm}\label{t:chi}
Let $w \in \bn$ \avoidingp{} have type-$\msfBC$ unit interval order $Q$,
%be $BC$-smooth with zig-zag network $Z = Z(w)$,
%avoid (what patterns?),
%let $B$ be the path matrix of
%let $Z = Z(w)$,
%let $P = P(w)$,
and fix bipartition $(\lambda,\mu) \vdash n$.
We have
%we may express irreducible characters in terms of
%the zig-zag network $Z = Z(w)$ as follows.
\begin{equation*}
(\chi\chi)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(Q,\lambda,\mu) \,|\,
U, V \text{ standard } \}.
\end{equation*}
\end{thm}
\begin{proof}
Let $A$ be the path matrix of $F_w$ and let $\chi^\lambda$, $\chi^\mu$
be the characters of $\mfs{k}$, $\mfs{n-k}$ satisfying
$(\chi\chi)^{\lambda,\mu} =
(\chi^\lambda \otimes \delta \chi^\mu) \upparrow_{\mfb{k,n-k}}^{\bn}$.
%($\star$ evaluation lemma?)
Lemma~\ref{l:tensorimm} and
we have
\begin{equation*}
(\chi\chi)^{\lambda,\mu}(\btc w1) = \bnimm{(\chi\chi)^{\lambda,\mu}}(A)
= \sumsb{I \subset [n]\\|I| = k}
\simm k{\chi^\lambda}(A_{I,I}^+) \simm{n-k}{\chi^\mu}(A_{[n] \ssm I, [n] \ssm I}^-).
\end{equation*}
Expanding irreducible character immanants
$\simm k{\chi^\lambda}$ and $\simm{n-k}{\chi^\mu}$
%in the sum
%of $\mfs{m}$ and $\mfs{n-m}$
in terms of induced trivial character immanants, we obtain
\begin{equation*}
%    \begin{aligned}
\sumsb{I \subset [n]\\|I| = k}
\sum_{\alpha \vdash k} K_{\alpha,\lambda}^{-1} \imm{\eta^\alpha}(A_{I,I}^+)
\ntksp \sum_{\beta \vdash n-k} \nTksp
K_{\beta,\mu}^{-1} \imm{\eta^\beta}(A_{[n]\ssm I,[n] \ssm I}^-).
\end{equation*}
Now let $Q$
be the type-$\msfBC$ unit interval order corresponding to $w$.
For any subset $J$ of $[n]$ and any partition $\nu \vdash |J|$,
let $r(Q_J,\nu)$ be the number of row-semistrict
$Q_J$-tableaux of shape $\nu$, and let $p(J)$
be the number of elements of $Q_J$ which are grounded.
%  and let $r^+(\gamma, J, Q)$ and $r^+(\gamma, J, Q)$
%  be the numbers of
%  row-semistrict $Q_J$-tableaux of shape $\gamma$
%  containing no grounded elements and
%  row-semistrict $Q_J$-tableaux of shape $\gamma$
%  containing any number of grounded elements, which may be circled.
By (\ref{eq:lmw}) and Proposition \ref{p:colstrictetc}, we have
%Corollary (need a $\lambda$-analog of Corollaries 7.11 - 7.13),
\begin{equation*}
\simm k{\eta^\alpha}(A_{I,I}^+) = 2^{p(I)} r(Q_I, \alpha),
\qquad
\end{equation*}
%    $ equals the number of
%  row-semistrict $Q_I$-tableaux of shape $\alpha$
%  containing any number of grounded elements, which may be circled,
\begin{equation*}
\simm{n-k}{\eta^\beta}(A_{[n]\ssm I,[n]\ssm I}^-) =
\begin{cases} r(Q_{[n]\ssm I}, \beta) &\text{if $p([n]\ssm I) = 0$}\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
%equals the number of
%row-semistrict $Q_{[n]\ssm I}$-tableaux of shape $\beta$
%containing no grounded elements.
Thus by \cite[Thm.\,4.7 (ii-b), (iii)]{CHSSkanEKL}, the sums
\begin{equation*}
\begin{gathered}
\sum_{\alpha \vdash k} K_{\alpha,\lambda}^{-1} \simm k{\eta^\alpha}(A_{I,I}^+)
= 2^{p(I)} \sum_{\alpha \vdash k} K_{\alpha,\lambda}^{-1} r(Q_I, \alpha),\\
\sum_{\beta \vdash n-k} \nTksp
K_{\beta,\mu}^{-1} \simm{n-k}{\eta^\beta}(A_{[n]\ssm I,[n]\ssm I}^-)
= \begin{cases}
\displaystyle{\sum_{\beta \vdash n-k} \nTksp
K_{\beta,\mu}^{-1} r(Q_{[n]\ssm I}, \beta)} &\text{if $p([n]\ssm I) = 0$},\\
0 &\text{otherwise}
\end{cases}
\end{gathered}
\end{equation*}
are equal to the numbers of
standard $Q_I$-tableaux of shape $\lambda$
containing any number of grounded elements which may be circled,
and standard $Q_{[n]\ssm I}$-tableaux of shape $\mu$
containing no grounded elements, respectively.
\end{proof}
For example, consider the evaluation $(\chi\chi)^{21,11}(\btc{2345\ol1}1)$
%applying Theorem~\ref{t:chi}
and the poset $Q(2345\ol1)$ in (\ref{eq:2111pex}).
%and the poset $bitableaux in (\ref{eq:2111pex}).
Of the bitableaux shown there, only the second contributes to this evaluation.

\begin{thm}\label{t:psi}
Let $w \in \bn$ \avoidingp{}
have type-$\msfBC$ unit interval order $Q$,
%  have zig-zag network $F_w$ and
%  Let $w \in \bn$ be $\msfBC$-smooth,
and fix bipartition
$(\lambda,\mu) \vdash n$.
%= ((\lambda_1,\dotsc,\lambda_r),(\mu_1,\dotsc,\mu_s))
%  of $n$, and define the sets $\mathcal S_1$, $\mathcal S_2$
%  of tableau properties
%  \begin{enumerate}[1.]
%    \item row-semistrict of type $e$,
%    \item excedance-free of type $e$,
%    \item row-closed and left-row-strict.
%  \end{enumerate}
%  \begin{equation*}
%    \begin{gathered}
%      \mathcal S_1 = \{\text{cyclically row-semistrict of type $e$},
%                      \text{record-free of type $e$},
%                      \text{cylindrical} \},\\
%      \mathcal S_2 = \{\text{left-anchored and row-semistrict of type $e$}\}.
%    \end{gathered}
%  \end{equation*}
%  Then we may express the evaluations at $\btc w1$ of
%  power sum characters in terms of
%  the zig-zag network $Z = Z(w)$ as follows.
%  \begin{enumerate}[1.]
%  \item
%  For any properties $i,j \in \mathcal S_1$, $k \in \mathcal S_2$ we have
We have
\begin{equation*}
%    \begin{aligned}
(\psi\psi)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(Q,\lambda,\mu) \,|\, U,V \text{ cyclically
row-semistrict } \}.
%      U \text{ has property } i, V \text{ has property } j \}\\
%      &= \lambda_1 \cdots \lambda_r \cdot
%      \# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\,
%      U \text{ has property } k, V \text{ has property } j \}\\
%      &= \mu_1 \cdots \mu_s \cdot
%      \# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\,
%      U \text{ has property } i, V \text{ has property } k \}\\
%      &= \lambda_1 \cdots \lambda_r \mu_1 \cdots \mu_s \cdot
%      \# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\,
%      U, V \text{ have property } k \}.
%      \end{aligned}
\end{equation*}
\end{thm}
\begin{proof}
%  ($\star$ Fix this: it is just a copy of the proof of the previous theorem.)
Let $A$ be the path matrix of $F_w$ and let $k = |\lambda|$.
%  $\psi^\lambda$, $\psi^\mu$
%  be the characters of $\mfs{m}$, $\mfs{n-m}$ satisfying
%  $(\psi\psi)^{\lambda,\mu} =
%  (\psi^\lambda \otimes \delta \psi^\mu) \upparrow_{\mfb{m,n-m}}^{\bn}$.
By Lemma~\ref{l:tensorimm} and
we have
\begin{equation}\label{eq:psipsisum}
(\psi\psi)^{\lambda,\mu}(\btc w1) = \bnimm{(\psi\psi)^{\lambda,\mu}}(A)
= \sumsb{I \subset [n]\\|I| = m}
\simm k{\psi^\lambda}(A_{I,I}^+) \simm{n-k}{\psi^\mu}(A_{[n] \ssm I, [n] \ssm I}^-).
\end{equation}
By \cite[Prop.\,2.4]{StemConj} we have
\begin{equation*}
\simm k{\psi^\lambda}(\bfx)
= \nTksp \sum_{(J_1,\dotsc,J_r)} \nTksp
\simm{\lambda_1}{\psi^{\lambda_1}}(\bfx_{J_1,J_1}) \cdots
\simm{\lambda_r}{\psi^{\lambda_r}}(\bfx_{J_r,J_r}).
\end{equation*}
Thus each term in the sum (\ref{eq:psipsisum}) is itself
a sum of products of type-$\msfA$ single-cycle
power sum trace immanants, evaluated at $A^+$ or $A^-$.
By Proposition~\ref{p:colstrictetc} each factor counts
one-row cyclically row-semistrict tableaux,
and their product counts
%cyclically row-semistrict
bitableaux of the required shape.
%($\star$ Mention bitableaux?)
\end{proof}

\begin{cor}\label{c:psi}
Let $w \in \bn$ \avoidingp{} have zig-zag network $F_w$ and
type-$\msfBC$ unit interval order $Q = Q(w)$.
In Theorem~\ref{t:psi}
we may replace
%  \{ (U,V) \in \bitab(Q,\lambda,\mu) \,|\,
%  U,V \text{ cyclically row-semistrict } \}
%  \end{equation*}
cyclically row-semistrict $Q$-tableaux
%shape $\nu$ and
%type $e$
with either of the following:
%column-strict $P$-tableaux.
%We may also replace
%row-semistrict $F_w$-tableaux of type $e$ with any of the following:
\begin{enumerate}%[(i)]
%\item cyclically row-semistrict $P$-tableaux,
%of shape $\nu$,
\item[(i)]
%$\{ (U,V) \in \bitab(Q,\lambda,\mu) \,|\,
%U,V \text{ record-free } \}$,
record-free $Q$-tableaux,
%shape $\nu$ and
%  type $e$,
%\item record-free $P$-tableaux,
%of shape $\nu$,
\item[(ii)]
%$\{ (U,V) \in \bitab(F_w,\lambda,\mu) \,|\,
%U,V \text{ cylindrical } \}$.
cylindrical $F_w$-tableaux.
%of shape $\nu$,
%\item disjoint cyclic vertex covers
%  of the directed graph $\ngr(Q)$ as defined
%  in \cite[\S 3.4]{SkanCCS}.
\end{enumerate}
We also have that $(\psi\psi)^{\lambda,\mu}(\btc w1)$ equals
\begin{enumerate}
\item[(iii)]
%    \begin{aligned}
%    \# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\, U,V \text{ cyclically
%      row-semistrict of type $e$} \}.
%      U \text{ has property } i, V \text{ has property } j \}\\
%      &= \lambda_1 \cdots \lambda_r \cdot
%      \# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\,
%      U \text{ has property } k, V \text{ has property } j \}\\
%      &= \mu_1 \cdots \mu_s \cdot
%      \# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\,
%      U \text{ has property } i, V \text{ has property } k \}\\
$       \lambda_1 \cdots \lambda_r \mu_1 \cdots \mu_s \cdot
\# \{ (U,V) \in \bitab(Q,\lambda,\mu) \,|\,
U, V \text{ left-anchored, row-semistrict } \}$.
%      \end{aligned}
%  \end{equation*}
%  and we may replace these $F_w$-tableaux with
%  left-anchored, row semistrict $P$-tableaux.
\item[(iv)]
the number of marked acyclic orientations of subgraph sequences
%      $(\inc(Q_{I_1}), \dotsc, \inc(Q_{I_r)), \inc(Q_{J_1}), \dotsc, \inc(Q_{J_s})))$,
%where $(I_1,\dotsc,I_r,J_1,\dotsc,J_s)}$
in which each oriented subgraph has one source,
and all grounded vertices appear in $I_1,\dotsc,I_r$.
%      is an ordered set partition of $[n]$ of type $(\lambda_1,\dotsc,\lambda_r,\mu_1,\dotsc,\mu_t)$.
\end{enumerate}
\end{cor}
\begin{proof}
By Proposition~\ref{p:colstrictetc},
one-rowed tableaux
in Theorem~\ref{t:psi}
which are cyclically
correspond bijectively to one-rowed tableaux
with properties
$(i)$ or $(ii)$
and to acyclic orientations $(iii)$ of
%the corresponding subgraphs of
$\inc(Q)$ which have one source.
Thus there is a correspondence of
several-rowed tableaux and of acyclic orientations of subgraph sequences.
%correspond as well
In all cases, we have equality of
the indices of paths, poset elements, and vertices which are grounded,
so markings correspond in the obvious way.
\end{proof}

For example, consider evaluating $(\psi\psi)^{21,11}(\btc{2345\ol1}1)$ by
applying Theorem~\ref{t:psi}
to the poset $Q(2345\ol1)$ in (\ref{eq:2111pex}).
Of the tableaux listed there, only the first and third contribute to
this trace evaluation.  If we modify the criteria of the theorem as in
Corollary~\ref{c:psi} (i), the same is true,
since a tableau with two or
fewer columns is $Q$-record-free if and only
if it is cyclically $Q$-row-semistrict.
On the other hand, if we modify the criteria of the theorem as in
Corollary~\ref{c:psi} (iii), then the first, second,
and fourth bitableaux contribute.

Now suppose that we modify the criteria of the theorem
as in Corollary~\ref{c:psi} (ii).  Then we may
apply these to path families
covering $F_{2345\ol1}$ in (\ref{eq:pathex}),
and we find that the first, third and fourth
tableaux in (\ref{eq:2111fex})
contribute to $(\psi\psi)^{21,11}(\btc{2345\ol1}1)$.

Finally, we may modify the criteria of the theorem
by applying Corollary~\ref{c:psi} (iii) to
%    $\inc(Q(2345\ol1))$
(\ref{eq:incQ23451}).  Both tableaux in (\ref{eq:markedor}) contribute,
since no component has more than one source.

%($\star$ Introduce this.)

%(Have Gavin do this?)
%Find a combinatorial interpretation for $\iota^{\lambda,\mu}(\btc w1)$
%when $w \in \bn$ \avoidsp.
%($\star$ Try to use \cite[Eqn.~(3.31)]{SkanCCS} and the appendix of
%[7]. Or just use the formula (\ref{eq:iotadef})
%and the corresponding immanant?)

%Show that if $w \in \bn$ \avoidsp, then there exists $v \in \bn$
%avoiding the signed patterns
%$\ol2\ol1$, $\ol21$, $1\ol2$, $312$, $3\ol12$,
%such that for all $\hbnq$-traces $\theta_q$ we have
%$\theta_q(\wtc wq) = \theta_q(\wtc vq)$.

%  Let $w \in \bn$ avoid patterns ?, ?, and let
%let $B$ be the path matrix of the zig-zag network $Z(w)$.
%Let $I$ be a subset of $[n]$.
%For $f(x)$ equal to each of the polynomials
%$\det(P(x)_{I,I})$, $\det(D(x)_{I,I})$, $\perm(P(x)_{I,I})$, $\perm(D(x)_{I,I})%$,
%find a combinatorial interpretation for $f(B)$.

%In each of these cases, does $f(B)$ count path tableaux with the properties
%  \begin{matrix}
%    \text{function}
%    &\text{shape}
%    &\begin{matrix}\text{row/col}\\ \text{condition} \end{matrix}
%    &\begin{matrix}\text{allow paths}\\ \text{in $[\ol a,a]$ star?} \end{matrix}\\
%    &  &  & \\
%  \det(P_{I,I})  & 1^k & \text{column strict} & \text{yes}\\
%  \det(D_{I,I})  & 1^k & \text{column strict} & \text{no}\\
%  \perm(P_{I,I}) & k   & \text{row semistrict} & \text{yes}\\
%  \perm(D_{I,I}) & k   & \text{row semistrict} & \text{no}
%  \end{matrix}
%or something similar?

%\bp Extend (modify?) the above interpretations so that they
%apply to an arbitrary type-$\msfBC$ star network $G$.
%(or zig-zag network or descending star network?),

%($\star$ Include power sum trace evaluations.)

Proposition~\ref{p:detpermpsiinterplong} -- Theorem~\ref{t:wtc1interpssubg}
have known $q$-analogs,
%and Proposition~\ref{p:approx} is stated in terms of $q$,
no such $q$-analogs are known for the results in Subsection~\ref{ss:mainBC}.
%are stated in terms of
%($\star$ Mention that it would be interesting to prove $q$-analogs of
%the above results.  Also give examples of computations which show that
%it is not easy to do so?)

\bp
State and prove $q$-analogs of the results in
Lemma~\ref{l:twobijections} --
%Theorem~\ref{t:epsiloneta} --
\ep

\section{Some equivalence relations}\label{s:equiv}

Given a Coxeter group $W$ and its Hecke algebra $H = H(W)$,
the trace space $\trsp(H)$ naturally partitions $H$ into equivalence classes
via the relation $\approx$ defined by
\begin{equation}\label{eq:approx}
D_1 \approx D_2 \text{ if for all } \theta_q \in \trsp(H) \text{ we have }
\theta_q(D_1) = \theta_q(D_2).
\end{equation}
%($\star$ Cite Geck and Pfeiffer (equality mod $[H,H]$).)
%(See, e.g., \cite[p.\,261]{GPCharHecke}.)
When $W = \sn$ or $\bn$, we may restrict this relation to the subset
of Kazhdan-Lusztig basis elements indexed by elements of $W$ \avoidingp.
Two more equivalence relations related to this restricted relation
are defined in terms of
isomorphism of the posets and graphs
described in Sections~\ref{s:uio} -- \ref{s:incgraph}:
%In particular
%we may define Kazhdan--Lusztig basis elements indexed by $v$, $w$
%to be equivalent if
$P(v) \cong P(w)$ ($Q(v) \cong Q(w)$), or $G(v) \cong G(w)$.
In type $\msfA$ these relations refine the first;
in types $\msfBC$, we conjecture the same to be true
and prove a weaker statement.

\ssec{Type-$\msfA$ equivalence relations}\label{ss:aequiv}

%and may restrict further to the subset of basis elements
%indexed by
%and further to another on the subset of basis elements indexed by
%\pavoiding elements of $W$.
%A different relation $\tripprox$  (or $\diffeo$?)
%on the Kazhdan--Lusztig basis elements indexed by
%\pavoiding elements of $W$ may be defined in terms
%of posets by
%  \wtc vq \ \tripprox \ \wtc wq \text{ if } P(v) \cong P(w).
%%if $P(v)$ and $P(w)$ are isomorphic as posets.
%Whether we define the equivalence relations in the type-$\msfA$ or type-$\msfBC$
%setting, we have that $\tripprox$ refines $\approx$
%($\star$ refer to theorems), i.e.,
%%$v \ \tripprox \ w$ implies $v \approx w$.
%that poset isomorphism $P(v) \cong P(w)$ implies equality of character
%evaluations $\chi(\wtc vq) = \chi(\wtc wq)$.

%($\star$ Rethink notation here?)

%\ssec{Type-$\msfA$ equivalence relations}

%($\star$ Remove references to future claims and introduce differently.)

The equivalence relation on
\begin{equation}\label{eq:smoothkl}
\{ \wtc wq \,|\, w \in \sn \text{ \avoidsp } \}
\end{equation}
defined by poset isomorphism $P(v) \cong P(w)$
refines that defined by graph isomorphism $G(v) \cong G(w)$,
which in turn refines
the restriction of the relation $\approx$ (\ref{eq:approx})
to this set.
%($\star$ Define decorated poset and graph isomorphism earlier.)
\begin{thm}\label{t:Aequivimplications}
For $v, w \in \sn$ \avoidingp, we have the implications
\begin{equation*}
P(v) \cong P(w)
\quad \Longrightarrow \quad
G(v) \cong G(w)
\quad \Longrightarrow \quad
\wtc vq \approx \wtc wq.
%    \theta_q(\wtc vq) = \theta_q(\wtc wq)
%    \text{ for all } \theta_q \in \trsp(\hnq).
\end{equation*}
\end{thm}
\begin{proof}
The first implication is clear; the second follows from
a $q$-analog of Theorem~\ref{t:wtc1interps}~(i-b) in
\cite[Prop.\,7.3 -- Thm.\,7.4]{CHSSkanEKL} which states that
all of the evaluations
$\{\epsilon_q^\lambda(\wtc wq) \,|\, \lambda \vdash n\}$
are determined by
\end{proof}
The converse of the first implication above is not true (\ref{eq:uioag});
the converse of the second is not known to be true.
\bp
For some $n$ find \pavoiding permutations $v, w \in \sn$
%for some $n$
which satisfy
$G(v) \not \cong G(w)$ and
%\quad \text{and} \quad
$\wtc vq \approx \wtc wq$
%  \theta_q(\wtc vq) = \theta_q(\wtc wq)
%  \text{ for all } \theta_q \in \trsp(\hnq),
%  \end{equation*}
or show that this is impossible.
\ep

%($\star$ Make sure this is open.
%Mention Orellana paper and others. Check with Jeremy Martin.)

By Theorems~\ref{t:312bij} and~\ref{t:Aequivimplications},
%and the correspondence $v \mapsto P(v)$
%of $312$-avoiding permutations in $\sn$
%and $n$-element unit interval orders (Theorem~\ref{t:312bij}),
%and the fact that trace evaluations at (\ref{eq:smoothkl}) depend
%only on the posets $P(w)$
%on $n$ elements,
%we see that
the problem of evaluating $\hnq$-traces at (\ref{eq:smoothkl})
reduces to the problem of evaluating these traces at the subset of
(\ref{eq:smoothkl}) indexed by $312$-avoiding
\begin{cor}\label{c:Acodominant}
For $w \in \sn$ \avoidingp, there exists $v \in \sn$ avoiding $312$
such that we have $\wtc wq \approx \wtc vq$.
%  $\theta_q(\wtc wq) = \theta_q(\wtc vq)$ for all $\theta_q \in \trsp(\hnq)$.
\end{cor}
Furthermore, some experimentation suggests that
the problem of evaluating traces
at {\em all} Kazhdan--Lusztig basis elements reduces to the
problem of evaluating traces at
the subset of (\ref{eq:smoothkl}) indexed by $312$-avoiding
With precise details of such a reduction not yet conjectured,
we have the following problem~\cite[Conj.\,1.9]{ANigroUpdate},
\cite[Conj.\,3.1]{HaimanHecke}.
\bp\label{prob:ANHcodom}
Show that for each $w \in \sn$ there exists a set $S = S(w) \subseteq \sn$
of $312$-avoiding permutations
and a set $\{p_{v,w}(q) \,|\, v \in S \} \subset \mathbb N[q]$
of polynomials such that we have
%for all $\theta_q \in \trsp(\hnq)$ we have
\begin{equation*}
\wtc wq \approx \sum_{v \in S} p_{v,w}(q) \wtc vq.
\end{equation*}
\ep
%($\star$ Mention that
We remark that Haiman~\cite[\S 3]{HaimanHecke}
introduced the name {\em codominant} for $312$-avoiding permutations
%for $w \in \sn$ avoiding the pattern $312$
because by (\ref{eq:laction}) these have the form
for $w_0$ the longest element of $\sn$ and
$v$ belonging to the dominant ($132$-avoiding) subset of
the vexillary ($2143$-avoiding) permutations defined in [42].

%By Proposition~\ref{p:anetpavoid}
%and Subsection~\ref{ss:auio},
%the $312$-avoiding permutations included in
%Proposition~\ref{p:detpermpsiinterp} and
%Theorems~\ref{t:wtc1interps} -- \ref{t:wtc1interpssubg}
%have a special relationship to the other permutations appearing there.
%Define an equivalence relation $\approx$ on the set
%  \begin{equation*}
%    \{w \in \sn \,|\, w \text{ \avoidsp} \}
%    \end{equation*}
%  by $v \approx w$ if $P(v) \cong P(w)$.
%  Then we have the following.
%  \begin{enumerate}
%  \item Each equivalence class of $\approx$ contains
%    exactly one representative which avoids the pattern $312$.
%  \item For each trace $\theta_q \in \trsp(\hnq)$
%    and each pair $v,w \in \sn$ \avoidingp, we have
%    $\theta_q(\wtc vq) = \theta_q(\wtc wq)$ if $v \approx w$.
%  \end{enumerate}
%Thus one can understand all trace evaluations of the form
%$\theta_q(\wtc wq) $ with $w$ \avoidingp{}
%by restricting attention to
%$w$ avoiding the pattern $312$.

%$\star$ Mention Haiman and Abreu--Nigro conjectures.

\ssec{Type-$\msfBC$ equivalence relations}

We distinguish between the $\msfBC$-analog
of the equivalence relation (\ref{eq:approx})
and that of its $q=1$ specialization.
For $D_1, D_2 \in \hbnq$, define
\begin{equation}\label{eq:approxbcq}
%  \begin{alignedat}{2}
D_1 \approx_q D_2 \text{ if for all } \theta_q \in \trsp(\hbnq)
\text{ we have } \theta_q(D_1) = \theta_q(D_2).
\end{equation}
For $D_1, D_2 \in \zbn$,
%and its $q=1$ specialization,
%  \text{ For $D_1, D_2 \in \zbn$,}
%D_1 \approx_1 D_2 \text{ if for all } \theta \in \trsp(\bn)
let $D_1 \approx_1 D_2$ be the $q=1$ specialization of the above.
%\text{ if for all } \theta \in \trsp(\bn)
%  \text{ we have } \theta(D_1) = \theta(D_2).
These relations naturally restrict to the Kazhdan--Lusztig bases of
$\hbnq$ and $\zbn$, and to the subsets of these indexed by elements $w \in \bn$
\avoidingp.
It is easy to show that the equivalence relation on
\begin{equation}\label{eq:smoothbc}
\{ \btc wq \,|\, w \in \bn\ {\text \avoidingp } \}
\end{equation}
defined by type-$\msfBC$ unit interval order
%(decorated) poset
isomorphism $Q(v) \cong Q(w)$
refines that defined by
type-$\msfBC$ indifference graph
%(decorated) graph
$\Gamma(v) \cong \Gamma(w)$,
which in turn refines the relation $\approx_1$.
\begin{thm}\label{t:BCequivimplications}
For $v, w \in \bn$ \avoidingp, we have the implications
\begin{equation*}
Q(v) \cong Q(w)
\quad \Longrightarrow \quad
\Gamma(v) \cong \Gamma(w)
\quad \Longrightarrow \quad
\btc v1 \approx_1 \btc w1.
%    \theta_q(\wtc vq) = \theta_q(\wtc wq)
%    \text{ for all } \theta_q \in \trsp(\hnq).
\end{equation*}
\end{thm}
\begin{proof}
The first implication is clear; the second follows from
\end{proof}
The converse of the first implication above is not true (\ref{eq:uiocg});
the converse of the second is not known to be true.
We conjecture that the second implication can be strengthened.
\begin{conj}\label{conj:BCequivimplications}
For $v, w \in \bn$ \avoidingp, we have the implication
%.  Then we have the implications
%  \begin{equation*}
$\Gamma(v) \cong \Gamma(w) \ \Longrightarrow \
%    \quad \Longrightarrow \quad
%    implies $
\btc vq \approx_q \btc wq$.
\end{conj}
The converse of this conjectured implication is not known to be true.

\bp
For some $n$ find \pavoiding permutations $v, w \in \bn$
%for some $n$
which satisfy
$\Gamma(v) \not \cong \Gamma(w)$
%\quad \text{and} \quad
and $\btc vq \approx_q \btc wq$,
%  \theta_q(\wtc vq) = \theta_q(\wtc wq)
%  \text{ for all } \theta_q \in \trsp(\hnq),
%  \end{equation*}
or show that this is impossible.
%($\star$ or restate this?)
\ep

%($\star$ Mention that the type-$\msfBC$ analog cannot be proved using the trace property
%$\chi(DD') = \chi(D'D)$ and commuting KL basis elements?)

By Theorems~\ref{t:dnetbcpavoid} and \ref{t:BCequivimplications},
%($\star$ A partial analog of Corollary~\ref{c:codominant} asserts that ?)
%($\star$ Add some words here?)
the problem of evaluating $\bn$-traces at
\begin{equation}\label{eq:smoothklbc}
\{ \btc w1 \,|\, w \in \bn \ {\text \avoidsp} \}
\end{equation}
reduces to the problem of evaluating these traces at the subset of
(\ref{eq:smoothklbc}) indexed by
elements of $\bn$ \avoidingsignedp.

%By Theorems~\ref{t:znetbcpavoid} -- \ref{t:dnetbcpavoid},
%the elements in $\bn$
%in Theorem~\ref{t:epsiloneta} -- Corollary~\ref{c:psi}
%have a special relationship
%to the other $\bn$-elements appearing there. In particular, we have the

%Define a relation
%The relation $\approx$ (\ref{eq:approx}) on \pavoiding elements of $\snn$
%  restricts to a relation $\approx'$
%  on \pavoiding elements of $\bn$.

%($\star$ Fix the following so that it applies to K--L basis elements, not just their indices.  Also, make sure it is correct.)
\begin{cor}\label{c:BCcodominant}
For $w \in \bn$ \avoidingp, there exists $v \in \bn$ \avoidingsignedp{}
such that we have $\btc w1 \approx_1 \btc v1$.
%  $\theta_q(\wtc wq) = \theta_q(\wtc vq)$ for all $\theta_q \in \trsp(\hnq)$.
\end{cor}
%Of course
If Conjecture~\ref{conj:BCequivimplications} is true, then the conclusion
of Corollary~\ref{c:BCcodominant} becomes $\btc wq \approx_q \btc vq$.
It would be interesting to discover the extent to which trace evaluations at
\begin{equation*}
\{ \btc wq \,|\, w \in \bn \ {\text \avoidssignedp} \}
\end{equation*}
describe trace evaluations at the entire Kazhdan--Lusztig basis of $\hbnq$,
as in Problem~\ref{prob:ANHcodom}.
One might call the above $\bn$-elements
{\em %$\msfBC$-
in analogy to codominant
permutations in $\sn$,
although the author is not aware of a definition of
dominant elements of $\bn$ in the literature.
On the other hand,
it would be interesting to relate
codominant elements of $\bn$
to the subsets of
leading, and
elements of $\bn$, which
%whose definitions
appear in
%appearing in
%which are defined in the literature,
%such as
%of $\bn$
%but we will not do this here.

%``Degeneracy locus formulas for amenable Weyl group elements'' on the arXiv.)

\section{Symmetric functions}\label{s:symm}

%($\star$ Rewrite this with better notation: right now there are too many H's?)
%(Try $\mathscr H$?)

For $W = \sn$ or $\bn$, and $H = H(W)$ its Hecke algebra,
bases of $\trsp(W)$ and $\trsp(H(W))$
are often studied in conjunction
with bases of an appropriate
%$\mathbb Z$- or $\zqq$-
module $\Lambda$
of symmetric functions.
When $W = \sn$, the symmetric function bases consist of
traces of the corresponding Lie group; when $W = \bn$ they do not.
In either case, we have the equalities
%in rank of the three modules
\begin{equation*}
\rnk(\trsp(W)) = \rnk(\trsp(H(W))) = \rnk(\Lambda),
\end{equation*}
which makes $\Lambda$ a convenient setting
in which to define generating functions for trace evaluations.

%For any group $G$, let $\trsp(G)$ be the $\mathbb Z$-module of
%{\em $G$-traces},
%linear functionals $\theta: G \rightarrow \mathbb Z$ satisfying
%$\theta(gh) = \theta(hg)$ for all $g, h \in \mathbb ZG$.
%For finite $G$, the rank of $\trsp(G)$
%is equal to the number of conjugacy classes of $G$.

%When $G = \sn$, this is the number of
%Define a
%It is well known that the rank of $\trsp(\sn)$ is the number of
\ssec{Type-$\msfA$ symmetric functions}

%Conjugacy classes of $\sn$ correspond to {\em integer partitions of $n$},
%weakly decreasing positive integer sequences
%$\lambda  = (\lambda_1,\dotsc,\lambda_\ell)$ satisfying
%$\lambda_1 + \cdots + \lambda_\ell = n$.
%The $\ell = \ell(\lambda)$ components of a partition $\lambda$
%are called its {\em parts}
%and we let the expressions
%$|\lambda| = n$ and $\lambda \vdash n$ denote that $\lambda$ is
%a partition of $n$.  Given $\lambda \vdash n$, we define the {\em transpose}
%partition $\lambda^\tr = (\lambda_1^\tr, \dotsc,\lambda_{\lambda_1}^{\tr})$ by
%$\lambda_i^\tr = \# \{ j \,|\, \lambda_j \geq i \}$.
%We define the empty sequence $\emptyset$ to be the unique partition of the
%integer $0$.
%The conjugacy class of $\sn$ corresponding to $\lambda \vdash n$
%is precisely the set of permutations
%whose {\em cycle type}, the weakly decreasing sequence of its
%cycle cardinalities, is equal to $\lambda$.

Corresponding to the six commonly used bases
of $\trsp(\hnq)$ and $\trsp(\sn)$
(\S \ref{ss:snhnq})
%correspond to standard bases
%ose used in the study
are six bases of the $\mathbb Z$-module
\begin{equation*}
\Lambda_n(x) = \Lambda_n(x_1, x_2, \dotsc ),
\end{equation*}
of homogeneous degree-$n$ symmetric functions:
the Schur basis $\{ s_\lambda \,|\, \lambda \vdash n\}$,
elementary basis $\{ e_\lambda \,|\, \lambda \vdash n \}$,
(complete) homogeneous basis $\{ h_\lambda \,|\, \lambda \vdash n \}$,
power sum basis $\{ p_\lambda \,|\, \lambda \vdash n\}$,
monomial basis $\{ m_\lambda \,|\, \lambda \vdash n \}$,
and forgotten basis $\{ f_\lambda \,|\, \lambda \vdash n \}$.
(See \cite[Ch.\,6]{StanEC2}.)
%($\star$ Define these here?)
The correspondence of trace bases and symmetric function bases
is given explicitly by (the $q$-extension of) the Frobenius map
\begin{align}
%  \frobch: \trsp(\sn) &\rightarrow \Lambda_n(x)\label{eq:frobch0}\\
\frobch_q: \trsp(\hnq) &\rightarrow \Lambda_n(x)\label{eq:frobch0}\\
\frobch_q(\theta_q)
&\defeq \frac1{n!}\sum_{w \in \sn} \theta(w)p_{\ctype(w)}\label{eq:frobch1}\\
&= \sum_{\mu \vdash n}  \frac1{z_\mu} \theta(\mu)p_\mu,\label{eq:frobch2}
\end{align}
where $\theta = \theta_1$ as in Section~\ref{s:trace},
and $\theta(\mu) \defeq \theta(w)$ for any $w \in \sn$ of cycle type $\mu$.
%and where $p_\lambda$ is the {\em power sum} symmetric function.
%and by the $q$-Frobenius map
%its $q$-analog
%  \begin{gathered}
%%  \begin{aligned}
%\frobch_q: \trsp(\hnq) \rightarrow \Lambda_n(x)\\
%%defined by
%\frobch_q(\theta_q) := \frobch(\theta), \qquad \qp12 \mapsto 1.
%  \end{gathered}
%  \end{equation*}
%($\star$ Are the scalars just integers?
%Or should we add $\qp12 \mapsto \qp12$?
%Or since we are just defining a correspondence of bases, doesn't it matter?
%Or should we add the words ``linear extension''?)
%to standard bases
%ose used in the study
%of the module $\Lambda_n$ of homogeneous degree-$n$ symmetric functions.
%(See \cite[Ch.\,6]{StanEC2}.)
%Three of the commonly used bases of $\trsp(\sn)$ consist of $\sn$-characters.
%These are the
%irreducible characters $\{ \chi^\lambda \,|\, \lambda \vdash n \}$,
%induced sign characters $\{ \epsilon^\lambda \,|\, \lambda \vdash n \}$,
%and induced trivial characters $\{ \eta^\lambda \,|\, \lambda \vdash n \}$,
%%  \begin{gathered}
%  \epsilon^\lambda = \sgn \upparrow_{\mfs\lambda}^{\sn},
%  \qquad \sgn(s_i) = -1,
%  \qquad \eta^\lambda = \triv \upparrow_{\mfs\lambda}^{\sn},
%  \qquad \triv(s_i) = 1,
%%  \end{gathered}
%and $\mfs{\lambda}$ is the Young subgroup of $\sn$ generated by
%  \{ s_1, \dotsc, s_{n-1} \} \ssm
%  \{ s_{\lambda_1}, s_{\lambda_1 + \lambda_2}, \dotsc, s_{n-\lambda_\ell} \}.
%These character bases
%%of $\trsp(\sn)$
%correspond to the
%Schur $\{ s_\lambda \,|\, \lambda \vdash n\}$,
%elementary $\{ e_\lambda \,|\, \lambda \vdash n \}$,
%and (complete) homogeneous $\{ h_\lambda \,|\, \lambda \vdash n \}$
%bases of $\Lambda_n$,
%in the sense that
Specifically, we have
\begin{equation}\label{eq:frobchs}
\begin{gathered}
\frobch_q(\chi_q^\lambda) = s_\lambda,
\qquad \frobch_q(\epsilon_q^\lambda) = e_\lambda,
\qquad \frobch_q(\eta_q^\lambda) = h_\lambda,\\
\qquad \frobch_q(\psi_q^\lambda) = p_\lambda,
\qquad \frobch_q(\phi_q^\lambda) = m_\lambda,
\qquad \frobch_q(\gamma_q^\lambda) = f_\lambda.
\end{gathered}
\end{equation}
%and are related to one another just as their Frobenius images are:
%as Schur, elementary, and homogenous symmetric functions are related:
%  \begin{alignedat}{2}
%    \epsilon^\lambda &= \sum_{\mu \vdash n} K_{\mu^{\tr}, \lambda} \chi^\mu,
%    &\qquad\qquad \eta^\lambda &= \sum_{\mu \vdash n} K_{\mu,\lambda} \chi^\mu,\\
%    e_\lambda &= \sum_{\mu \vdash n} K_{\mu^{\tr}, \lambda} s_\mu,
%    &\qquad\qquad h_\lambda &= \sum_{\mu \vdash n} K_{\mu,\lambda} s_\mu,
%  \end{alignedat}
%where $\{K_{\lambda,\mu} \,|\, \lambda, \mu \vdash n \}$ are
%the {\em Kostka numbers}.

We construct generating functions for $\hnq$-trace evaluations as follows.
Given element $D \in \mathbb Q(q) \otimes \hnq$, define
%we can define
the generating function
\begin{equation*}
Y_q(D) \defeq \sum_{\lambda \vdash n} \epsilon_q^\lambda(D) m_\lambda
%  \in \zqq \otimes \Lambda_n(x)
\in \mathbb Q(q) \otimes \Lambda_n(x)
\end{equation*}
for the evaluation of induced sign characters at $D$.
By \cite[Prop.\,2.1]{SkanCCS}, this symmetric function is in fact a generating
function for the evaluation of {\em all} the standard traces at $D$, because it
is equal to
\begin{equation*}
\sum_{\lambda \vdash n} \eta_q^\lambda(D)f_\lambda
= \sum_{\lambda \vdash n}\frac{(-1)^{n-\ell(\lambda)}\psi_q^\lambda(D)}{z_\lambda}p_\lambda
= \sum_{\lambda \vdash n} \chi_q^{\lambda^\tr}(D)s_\lambda
= \sum_{\lambda \vdash n} \phi_q^\lambda(D)e_\lambda
= \sum_{\lambda \vdash n} \gamma_q^\lambda(D)h_\lambda.
\end{equation*}
Equivalently, if we let
$\omega: \Lambda_n(x) \rightarrow \Lambda_n(x)$
be the standard involution mapping
\begin{equation}\label{eq:omega}
s_\lambda \mapsto s_{\lambda^\tr},
\qquad e_\lambda \mapsto h_\lambda,
\qquad p_\lambda \mapsto (-1)^{n - \ell(\lambda)}p_\lambda,
\qquad m_\lambda \mapsto f_\lambda,
\end{equation}
%(See \cite[Ch.\,6]{StanEC2}.)
then we have that
$\omega Y_q(D)$ is equal to
\begin{equation*}
\sum_{\lambda \vdash n} \epsilon_q^\lambda(D)f_\lambda
= \sum_{\lambda \vdash n} \eta_q^\lambda(D)m_\lambda
= \sum_{\lambda \vdash n}\frac{\psi_q^\lambda(D)}{z_\lambda}p_\lambda
= \sum_{\lambda \vdash n} \chi_q^\lambda(D)s_\lambda
= \sum_{\lambda \vdash n} \phi_q^\lambda(D)h_\lambda
= \sum_{\lambda \vdash n} \ntnsp \gamma_q^\lambda(D)e_\lambda.\nTksp
\end{equation*}
It is not difficult to show that
every symmetric function in $\mathbb Q(q) \otimes \Lambda_n(x)$
is $Y_q(D)$ for some $D \in \mathbb Q(q) \otimes \hnq$.
(See \cite[Prop.\,3]{SkanCCS}.)

Theorem~\ref{t:wtc1interps} ($i$-$b$)
shows that
%$g = \wtc wq$
%$ is a Kazhdan--Lusztig basis element $
$w \in \sn$ \avoidsp,
the symmetric function $Y_q(\wtc wq)$
is related to
%decompositions of a poset into chains.
%It is also closely related to
colorings of $\inc(P(w))$.
%an incomparability graph.
%($\star$ Mention Stanley chromatic symmetric function first.)
More generally,
Stanley~[58] defined
%for any simple graph $G = (V,E)$,
the {\em chromatic symmetric function} of any simple graph $G$ to be
\begin{equation}\label{eq:chrom}
X_G \defeq \sum_\kappa x_1^{|\kappa^{-1}(1)|} x_2^{|\kappa^{-1}(2)|} \cdots,
%  X_G \defeq \sum_\kappa x_{\kappa(1)} \cdots x_{\kappa(n)},
\end{equation}
where the sum is over all proper colorings
$\kappa: V \rightarrow \{1,2,\dotsc, \}$ of $G$ (\S \ref{ss:aincg}).
%as in Subsection~\ref{ss:Aposetgraph}.
%(A coloring $\kappa$ is called {\em proper} if $(a,b) \in E$ implies
%that $\kappa(a) \neq \kappa(b)$.)
%For the incomparability graph $\inc(P)$ of a poset $P$,
%we may express the symmetric function
%, we may equivalently express
%$X_{\inc(P)}$ in terms of
%decompositions of $P$ into chains.
Expanding in the monomial basis of $\Lambda_n$, we have
%Letting $c_{P,\lambda}$ be the number of column-strict $P$-tableaux of shape
%$\lambda^\tr$, we have
%Then we have
\begin{equation*}
X_G = \sum_\lambda c_{G,\lambda} m_\lambda,
\end{equation*}
where $c_{G,\lambda}$ is the number of
proper colorings of $G$ of type $\lambda$.

%($\star$ Decide the level of detail necessary for the Shareshian-Wachs
%chromatic quasisymmetric function.)

Shareshian and Wachs~[53] defined a quasisymmetric extension
$X_{G,q}$ of the
%Stanley's chromatic
symmetric function $X_G$.
%Let $G = (V,E)$ be a simple directed graph on vertices labeled $1, \dotsc, n$.
Given a proper coloring $\kappa$
%: V \rightarrow \{1,2,\dotsc, \}$
of $G$,
%a (simple undirected) graph $G = (V,E)$,
define $\inv_G(\kappa)$ to be the number of pairs $(i,j) \in E$ with
$i < j$ and $\kappa(i) > \kappa(j)$.
For any composition $\alpha = (\alpha_1, \dotsc, \alpha_\ell) \vDash n$, define
\begin{equation*}
c_{G,\alpha}(q) = \nTksp \sumsb{\kappa \text{ proper}\\ \type(\kappa) = \alpha} \nTksp
\end{equation*}
and let
\begin{equation*}
M_\alpha = \ntksp \sum_{i_1 < \cdots < i_\ell} \ntksp
x_{i_1}^{\alpha_1} \cdots x_{i_\ell}^{\alpha_\ell}
\end{equation*}
be the {\em monomial quasisymmetric function} indexed by $\alpha$.
Then we have the definition
%  X_{G,q} = \sum_{\alpha \vDash n} c(G,\alpha,q) M_\alpha.
\begin{equation}\label{eq:chromq}
X_{G,q} = \sum_\kappa q^{\inv_G(\kappa)}
x_1^{|\kappa^{-1}(1)|} x_2^{|\kappa^{-1}(2)|} \cdots
%x_{\kappa(1)} \cdots x_{\kappa(n)}
= \sum_{\alpha \vDash n} c_{G,\alpha}(q) M_\alpha,
\end{equation}
where the first sum is over
proper colorings of $G$.
It is easy to see that the $q=1$ specialization of $X_{G,q}$ satisfies
$X_{G,1} = X_G$.
%($\star$ Define chromatic quasisymmetric functions and state that
When $G = \inc(P)$ for a unit interval order $P$
labeled as in Algorithm \ref{a:ptow},
the quasisymmetric function $X_{\inc(P),q}$
is in fact symmetric \cite[Thm.\,4.5]{SWachsChromQF}.
Furthermore, we have the following~\cite[Thm.\,7.4]{CHSSkanEKL}.
\begin{thm}\label{t:YequalsX}
For $w \in \sn$ \avoidingp,
and $P = P(w)$ defined as in Subsection~\ref{ss:auio},
we have
$Y_q(\wtc wq) = X_{\inc(P),q}$.
\end{thm}
%where $P$ corresponds to $w$.)

%($\star$ Mention Hessenberg variety connection.)
%($\star$ Mention $q=1$ specialization.)

\ssec{Type-$\msfBC$ symmetric functions}

Corresponding to the eleven commonly used bases
of $\trsp(\hbnq)$ and $\trsp(\bn)$
(\S \ref{ss:bnhbnq})
%correspond to standard bases
%ose used in the study
are eleven natural bases of the $\mathbb Z$-module
%  \Lambda_n(x) = \Lambda_n(x_1, x_2, \dotsc ),
%of homogeneous degree-$n$ symmetric functions:
%($\star$ ``Turn around'' the wording to reflect the fact that
%traces were defined earlier, and symmetric functions are being defined now.)
%Eleven bases of $\trsp(\bn)$
%correspond to natural bases
%ose used in the study
%of the module
of {\em type-$\msfBC$ symmetric functions} of degree $n$,
\begin{equation*}
\Lambda_n(x,y) \defeq \bigoplus_{k=0}^n \Lambda_k(x) \otimes \Lambda_{n-k}(y),
\end{equation*}
where $x = (x_1, x_2, \dotsc )$ and $y = (y_1, y_2, \dotsc )$.
These bases of $\Lambda_n(x,y)$ consist
of ten {\em nonplethystic bases}
of the form
$(og)_{\lambda,\mu} \defeq o_\lambda(x)g_\mu(y)$,
\begin{equation}\label{eq:bcnpbases}
\begin{gathered}
\{(ss)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
\{(ee)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
\{(hh)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \\
\{(eh)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
\{(he)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \\
\{(mm)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
\{(f\ntnsp f)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
\{(mf)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \\
\{(fm)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
\{(pp)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \\
%\{s_\lambda(x)s_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{e_\lambda(x)e_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{e_\lambda(x)h_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \\
%\{h_\lambda(x)e_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{h_\lambda(x)h_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \\
%\{m_\lambda(x)m_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{m_\lambda(x)f_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{f_\lambda(x)m_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \\
%\{f_\lambda(x)f_\mu(y)\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{(ss)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \}, \quad
%\{(pp)_{\lambda,\mu}\,|\, (\lambda,\mu) \vdash n \},
%\{p^+_\lambda(x,y)p^-_\mu(x,y)\,|\, (\lambda,\mu) \vdash n \},
\end{gathered}
\end{equation}
%  (bc)_{\lambda,\mu} \defeq b_\lambda(x)c_\mu(y),
and the {\em plethystic power sum basis}
\begin{equation}\label{eq:pbasis}
%  \{p^+_\lambda p^-_\mu\,|\, (\lambda,\mu) \vdash n \},
\{p^+_\lambda p^-_\mu\,|\, (\lambda,\mu) \vdash n \},
\end{equation}
%where $p^+_\lambda$, $p^-_\mu$ are
defined in terms of
ordinary power sum symmetric functions
$p_k(x) \defeq x_1^k + x_2^k + \cdots$ by
\begin{equation}\label{eq:pbasis2}
\begin{gathered}
p_\lambda^+ \defeq
%p_\lambda[X+Y] =
(p_{\lambda_1}(x) + p_{\lambda_1}(y)) \cdots (p_{\lambda_{\ell(\lambda)}}(x) + p_{\lambda_{\ell(\lambda)}}(y)),\\
p_\mu^- \defeq
%= p_\mu[X-Y] =
(p_{\mu_1}(x) - p_{\mu_1}(y)) \cdots (p_{\mu_{\ell(\mu)}}(x) - p_{\mu_{\ell(\mu)}}(y)).
\end{gathered}
\end{equation}
The functions (\ref{eq:pbasis2})
%$p_\lambda^+$, $p_\mu^-$
often appear in the literature as
%aswritten in plethystic notation
$p_\lambda[X+Y], p_\mu[X-Y]$, respectively.
A correspondence between bases of $\trsp(\hbnq)$ and the bases
(\ref{eq:bcnpbases}) -- (\ref{eq:pbasis}) of
$\Lambda_n(x,y)$ is given
explicitly by the ($q$-extension of the)
plethystic $\msfBC$-Frobenius map~\cite[\S 1, App.~B]{M1}.
(See also~\cite[Eq.\,(2.5)]{AAERCharFormulasHyper}.)
%Adin et al.
%define the type-$B$
%Frobenius characteristic map by
%In particular
%, for $(\lambda, \mu) \vdash n$ and $\lambda \vdash a$,
%he defines
\begin{align}
%  \begin{aligned}
\pfrobch_q\ntksp:
\trsp(\hbnq) &\rightarrow \Lambda_n(x,y) \label{eq:frobchaaer0}\\
%    \bigoplus_{a=0}^n \Lambda_a(x) \otimes \Lambda_{n-a}(y)\\
%    \iota^{\lambda,\mu} &\mapsto
%    \frac{p_\lambda(x+y) p_\mu(x-y)}{z_\lambda z_\mu 2^{\ell(\lambda) + \ell(\mu)}}.
\pfrobch_q(\theta_q) &= \frac1{2^nn!} \ntnsp \sum_{w \in \bn} \ntksp \theta(w)
p^+_{\alpha(w)} p^-_{\beta(w)} \label{eq:frobchaaer1} \\
&= \nTksp \sum_{(\lambda,\mu) \vdash n} \ntnsp
%  \frac{\theta(\alpha,\beta)}
\frac{\theta(\lambda,\mu)}{z_\lambda z_\mu 2^{\ell(\lambda)+\ell(\mu)}}
p^+_{\lambda} p^-_{\mu}, \label{eq:frobchaaer2}
%  \end{aligned}
\end{align}
where $\alpha(w)$, $\beta(w)$ are the partitions satisfying
$\sct(w) = (\alpha(w), \beta(w))$, and where we define
$\theta(\lambda,\mu) \defeq \theta(w)$ for any $w \in \bn$
having $\sct(w) = (\lambda,\mu)$.
%  \end{aligned}
%When the bases in (\ref{eq:basispair}) are character bases,
%the definition (\ref{eq:inducedpair}) gives a character basis of
%we have
%Examples are the
%irreducible and induced one-dimensional characters,
%which correspond to the
%Schur and elementary-homogeneous
%bases (\ref{eq:bcnpbases}) of $\Lambda_n(x,y)$ by
%$\{ s_\lambda(x)s_\lambda(y) \,|\, (\lambda,\mu) \vdash n\}$,
%elementary $\{ e_\lambda \,|\, \lambda \vdash n \}$,
%and (complete) homogeneous $\{ h_\lambda \,|\, \lambda \vdash n \}$
%bases of $\Lambda_n$,
%in the sense that
Specifically, $\pfrobch_q$ maps
\begin{equation}\label{eq:pfrobchtable}
\begin{gathered}
%  \qquad \pfrobch((\epsilon\epsilon)^{\lambda,\mu}) = e_\lambda(x)e_\mu(y),
%  \qquad \pfrobch((\epsilon\eta)^{\lambda,\mu}) = e_\lambda(x)h_\mu(y),
%  \qquad \pfrobch((\eta\epsilon)^{\lambda,\mu}) = h_\lambda(x)e_\mu(y),
%  \qquad \pfrobch((\eta\eta)^{\lambda,\mu}) = h_\lambda(x)h_\mu(y),
\quad (\eta\eta)^{\lambda,\mu}_q \mapsto (hh)_{\lambda,\mu},
\quad (\eta\epsilon)^{\lambda,\mu}_q \mapsto (he)_{\lambda,\mu},
\quad (\epsilon\eta)^{\lambda,\mu}_q \mapsto (eh)_{\lambda,\mu},
\quad (\epsilon\epsilon)^{\lambda,\mu}_q \mapsto (ee)_{\lambda,\mu},
\\
%  \end{gathered}
%  \begin{gathered}
%  \pfrobch\ntksp:
(\phi\phi)^{\lambda,\mu}_q \mapsto (mm)_{\lambda,\mu},
\quad (\phi\gamma)^{\lambda,\mu}_q \mapsto (mf)_{\lambda,\mu},
\quad (\gamma\phi)^{\lambda,\mu}_q \mapsto (fm)_{\lambda,\mu},%\\
\quad (\gamma\gamma)^{\lambda,\mu}_q \mapsto (f\ntnsp f)_{\lambda,\mu},\nTksp\\
(\chi\chi)^{\lambda,\mu}_q \mapsto (ss)_{\lambda,\mu},
\quad (\psi\psi)^{\lambda,\mu}_q \mapsto (pp)_{\lambda,\mu},
\quad \iota^{\lambda,\mu}_q \mapsto p^+_\lambda p^-_\mu.
\end{gathered}
\end{equation}
We extend the involutive homomorphism $\omega$ on $\Lambda_n(x)$
to an involutive homomorphsim on $\Lambda_n(x,y)$ in the simplest way:
$\omega(o_\lambda(x)g_\mu(y)) \defeq \omega(o_\lambda(x))\omega(g_\mu(y))$.
%  \qquad
This exchanges the symmetric functions on line $1$ of
(\ref{eq:pfrobchtable}) with the corresponding functions on line $2$,
transposes the index shapes of $(ss)_{\lambda,\mu}$ and multiplies
each power sum basis element by $(-1)^{\ell(\lambda)+\ell(\mu)}$.
%($\star$ Cite this.)
%  \omega(ss)_{\lambda,\mu} = (ss)_{\lambda^\tr,\mu^\tr}, \qquad
%\omega(pp)_{\lambda,\mu} = (-1)^{\ell(\lambda) + \ell(\mu)}(pp)_{\lambda,\mu}, %\qquad
%%= \omega(p_\lambda^+)\omega(p_\mu^-)
%= (-1)^{\ell(\lambda) + \ell(\mu)}
%p_\lambda^+ p_\mu^-.
%($\star$ Mention $\omega$ here, or somewhere.)
Transition matrices relating the ten nonplethystic bases
have entries which are simply products of entries of transition
matrices relating type-$\msfA$ symmetric functions, e.g.,
\begin{equation}\label{eq:sstomm}
(ss)_{\lambda,\mu} = \nTksp \sum_{(\alpha,\beta) \vdash n} \nTksp
K_{\lambda, \alpha}K_{\mu, \beta}  (mm)_{\alpha,\beta}.
\end{equation}
%($\star$ Mention the transition matrices with products of Kostka numbers here,
%or earlier.)
The plethystic power sum basis can be related to the others via the
nonplethystic Schur basis,
\begin{equation}\label{eq:pptoss}
p_\lambda^+p_\mu^-  = \nTksp \sum_{(\alpha,\beta)\vdash n} \nTksp
(\chi\chi)^{\alpha,\beta}(\lambda,\mu) (ss)_{\alpha,\beta}.
\end{equation}

We construct generating functions for $\hbnq$-trace evaluations as follows.
Given any element $D \in \hbnq$, define the generating function
%{\em type-$\msfBC$ symmetric generating function}
\begin{equation}\label{eq:YBCdefn}
\YBCq(D) = \sum_{\lambda \vdash n}
(\epsilon\epsilon)^{\lambda,\mu}_q(D) (mm)_{\lambda,\mu}
\in \zqq \otimes \Lambda_n(x,y).
\end{equation}
%($\star$ Change the name to $Y^{\msfBC}$ or $Y^{\mathfrak B}$, or something similar?)
%($\star$ Define a $q$-version of this too.)
%($\star$ Define a similar function
%  \sum_{\lambda \vdash n} (\epsilon\epsilon)^{\lambda,\mu}(g) m^+_\lambda m^-_\mu
%  \in \Lambda_n(x,y)
%and its $q$-analog in terms of plethystic symmetric
%function bases, and give it a different name?)
This symmetric function is in fact a generating
function for the evaluation of {\em all} the standard traces at $D$,
in the following sense.
\begin{prop}\label{p:YBCexpansions}
For $D \in \hbnq$ we have
\begin{equation*}
\begin{aligned}
\YBCq(D)
&= \sum_{(\lambda,\mu)\vdash n} (\epsilon\epsilon)^{\lambda,\mu}_q(D) (mm)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\epsilon\eta)^{\lambda,\mu}_q(D) (mf)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\eta\epsilon)^{\lambda,\mu}_q(D) (fm)_{\lambda,\mu}\\
&= \sum_{(\lambda,\mu)\vdash n} (\eta\eta)^{\lambda,\mu}_q(D) (f\ntnsp f)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\chi\chi)^{\lambda^\tr,\mu^\tr}_q(D) (ss)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\phi\phi)^{\lambda,\mu}_q(D) (ee)_{\lambda,\mu}\\
&= \sum_{(\lambda,\mu)\vdash n} (\phi\gamma)^{\lambda,\mu}_q(D) (eh)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\gamma\phi)^{\lambda,\mu}_q(D) (he)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\gamma\gamma)^{\lambda,\mu}_q(D) (hh)_{\lambda,\mu}\\
&= \sum_{(\lambda,\mu)\vdash n}
\frac{(-1)^{\ell(\lambda)+\ell(\mu)}(\psi\psi)^{\lambda,\mu}_q(D)}{z_\lambda z_\mu}
= \sum_{(\lambda,\mu)\vdash n}
(-1)^{\ell(\lambda)+\ell(\mu)} \iota^{\lambda,\mu}_q(D) p^+_\lambda p^-_\mu.
\end{aligned}
\end{equation*}
%  ($\star$ Is the coefficient of $p^+_\lambda p^-_\mu$ correct?)
Equivalently, $\omega\YBCq(D)$
%($\star$ Define $\omega$ earlier.)
is equal to
\begin{equation}\label{eq:omegabc}
\begin{aligned}
&\sum_{(\lambda,\mu)\vdash n} (\epsilon\epsilon)^{\lambda,\mu}_q(D) (f\ntnsp f)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\epsilon\eta)^{\lambda,\mu}_q(D) (fm)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\eta\epsilon)^{\lambda,\mu}_q(D) (mf)_{\lambda,\mu}\\
&= \sum_{(\lambda,\mu)\vdash n} (\eta\eta)^{\lambda,\mu}_q(D) (mm)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\chi\chi)^{\lambda,\mu}_q(D) (ss)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\phi\phi)^{\lambda,\mu}_q(D) (hh)_{\lambda,\mu}\\
&= \sum_{(\lambda,\mu)\vdash n} (\phi\gamma)^{\lambda,\mu}_q(D) (he)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\gamma\phi)^{\lambda,\mu}_q(D) (eh)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} (\gamma\gamma)^{\lambda,\mu}_q(D) (ee)_{\lambda,\mu}\\
&= \sum_{(\lambda,\mu)\vdash n} \frac{(\psi\psi)^{\lambda,\mu}_q(D)}{z_\lambda z_\mu} (pp)_{\lambda,\mu}
= \sum_{(\lambda,\mu)\vdash n} \iota^{\lambda,\mu}_q(D) p^+_\lambda p^-_\mu.
\end{aligned}
\end{equation}
%($\star$ Is the coefficient of $p^+_\lambda p^-_\mu$ correct?)
\end{prop}
\begin{proof}
Consider the fourth and fifth sums in (\ref{eq:omegabc}),
in which the symmetric functions and traces satisfy
%  Since we have
\begin{equation}\label{eq:kostkasum}
(ss)_{\lambda,\mu} = \nTksp \sum_{(\alpha,\beta) \vdash n} \nTksp
K_{\lambda, \alpha}K_{\mu, \beta}  (mm)_{\alpha,\beta},
\qquad
(\eta\eta)^{\alpha,\beta}_q = \nTksp \sum_{(\lambda,\mu) \vdash n} \nTksp
K_{\lambda, \alpha}K_{\mu, \beta} (\chi\chi)^{\lambda,\mu}_q.
\end{equation}
Using (\ref{eq:kostkasum}) to expand the fifth sum in the
monomial symmetric function basis, we have
\begin{equation*}
\begin{aligned}
\sum_{(\lambda,\mu)\vdash n} \nTksp (\chi\chi)^{\lambda,\mu}_q(D)
\nTksp \sum_{(\alpha,\beta) \vdash n} \nTksp
K_{\lambda,\alpha}K_{\mu,\beta} (mm)_{\alpha,\beta}
&= \sum_{(\alpha,\beta) \vdash n} \sum_{(\lambda,\mu)\vdash n}
K_{\lambda,\alpha}K_{\mu,\beta} (\chi\chi)^{\lambda,\mu}_q(D) (mm)_{\alpha,\beta} \\
&= \sum_{(\alpha,\beta) \vdash n} \ntksp
(\eta\eta)^{\alpha,\beta}_q(D) (mm)_{\alpha,\beta},
\end{aligned}
\end{equation*}
i.e., it is equal to the fourth sum.
%    Now for the last statement.
%One can check (e.g., in \cite[\S 2]{RemTrans}) that
Similarly, for each of the remaining sums of the form
$\sum_{(\lambda,\mu) \vdash n} (\zeta\xi)^{\lambda,\mu}_q(D) o_\alpha(x) g_\beta(y)$
in (\ref{eq:omegabc}),
there is a matrix
$(M_{(\lambda,\mu),(\alpha,\beta)})_{(\lambda,\mu) \vdash n, (\alpha,\beta) \vdash n}$
and equations
\begin{equation*}
(ss)_{\lambda,\mu} = \sum_{(\alpha,\beta)}
M_{(\lambda,\mu),(\alpha,\beta)} o_\alpha(x)g_\beta(y),
\qquad
(\zeta\xi)^{\alpha,\beta}_q = \sum_{(\lambda,\mu)}
M_{(\lambda, \mu),(\alpha,\beta)} (\chi\chi)^{\lambda,\mu}_q,
\end{equation*}
relating it to the fifth sum.
In particular,
we have
$M_{(\lambda,\mu),(\alpha,\beta)} =
%    $\chi^\lambda(\mu)$,
%    These matrices are described in detail in \cite[\S 2]{RemTrans} using
%    the notation
%where coefficients $M_{\lambda,\mu}$ vary over matrices called
%    $(K_{\lambda^\tr,\mu})_{\lambda,\mu}$,
%    $(\chi^\lambda_\mu)_{\lambda,\mu}$,
%    $(K^{-1}_{\mu, \lambda})_{\lambda,\mu}$,
%    $(K^{-1}_{\mu, \lambda^\tr\;})_{\lambda,\mu}$,
%    where $\chi^\lambda(\mu)$ is the value $\chi^\lambda(w)$ for any
%    $w \in \sn$ having cycle type $\mu$.
(See \cite[\S 3]{RemTrans}.)
Relating the last sum to the fifth sum, we have equations
\begin{equation*}
(ss)_{\lambda,\mu} = \sum_{(\alpha,\beta)}
\frac{(\chi\chi)^{\lambda,\mu}(\alpha,\beta)}
{z_\alpha z_\beta 2^{\ell(\alpha) + \ell(\beta)}}
p^+_\alpha p^-_\beta,
\qquad
\iota^{\alpha,\beta} = \sum_{(\lambda,\mu)}
\frac{(\chi\chi)^{\lambda,\mu}(\alpha,\beta)}
{z_\alpha z_\beta 2^{\ell(\alpha) + \ell(\beta)}}
\end{equation*}
%    $\sum_{(\lambda,\mu)\vdash n} \iota^{\lambda,\mu}(D) p^+_\lambda p^-_\mu$
%    we have $M_{(\lambda,\mu),(\alpha,\beta)} = (\chi\chi)^{\lambda,\mu}(\alpha,\beta)$.
%   ($\star$ Is this last pair of equations correct?)
%    Probably not: needs powers of $2$ and $z_\lambda z_\beta$ or $z_\alpha z_\beta$.)
\end{proof}

To say that the functions $\{ Y_q(D) \,|\, D \in \zbn \}$ arise often
in the study of type-$\msfBC$ symmetric functions would be an
essentially {\em every} element of $\mathbb Z[q] \otimes \Lambda_n(x,y)$
has this form.

%($\star$ For $g \in \hbnq$, define
%  Y_q(D) = \sum_{(\lambda,\mu) \vdash n}
%  (\epsilon\epsilon)_q^{\lambda,\mu}(D)(mm)_{\lambda,\mu}
%somewhere earlier, and define $\hbnq$-traces earlier too.)
%($\star$ Is it useful to state and prove a $q$-analog of the following?
%Or should we just use the second $q=1$ proof?)

%$\star$ Define the trace space of the $\hbnq$ and the $q$-analogs of bases?
%  Y_q(D)  = \sum_{\lambda \vdash n} \epsilon_q^\lambda(D) m_\lambda ?
%or what would be the best notation for this?
%($\star$ Put this somewhere?)
%When $w$ \avoidsp~($w_1 \cdots w_n$ contains no subword
%$w_{i_1}w_{i_2}w_{i_3}w_{i_4}$ whose letters have values appearing
%in the same relative order as $4231$ or $3412$),
%each polynomial $P_{v,w}(q)$ is identically $1$.
%The universality of the generating functions $Y_q(D)$ may now be stated as follows.
\begin{prop}\label{p:everysymmfnbcq}
%  Let $Q$ be an integral domain and let $F$ be its field of fractions.
Every symmetric function in $\mathbb Z[q] \otimes \Lambda_n(x,y)$
has the form $Y_q(D)$ for some element
%$D \in \mathbb Q[\bn]$.
$D \in \mathbb Q(q)[\bn]$.
\end{prop}
\begin{proof}
%    Now consider the second statment.
Fix a symmetric function in $\Lambda_n(x,y)$
%$\mathbb Z[q] \otimes \Lambda_n(x,y)$,
and expand it in the plethystic power sum basis as
$\sum_{(\lambda,\mu) \vdash n} a_{\lambda,\mu}(q) p_\lambda^+p_\mu^-$.
%$\sum_{(\lambda,\mu) \vdash n} a_{\lambda,\mu} p_\lambda^+(x,y)p_\mu^-(x,y)$.
Then choose one representative $w_{\kappa,\nu}$
of each conjugacy class of $\bn$.
We have
\begin{equation*}
\iota^{\lambda,\mu}_q(T_{w_{\kappa,\nu}}) =
\sum_{(\alpha,\beta)\vdash n} (\chi\chi)^{\alpha,\beta}(\lambda,\mu)
\end{equation*}
%  ($\star$ Are any powers of $2$ or $z_\lambda$'s missing?)
Since the matrices
are both invertible, so is their product
Call the inverse of this product $B = (b_{(\alpha,\beta),(\lambda,\mu)}(q))$
and for each $(\alpha,\beta) \vdash n$ define
\begin{equation*}
U_{\alpha,\beta} =
\sum_{(\kappa,\nu)\vdash n} b_{(\alpha,\beta),(\kappa,\nu)}(q) T_{w_{\kappa,\nu}}
\in \mathbb Q(q) \otimes \hbnq.
\end{equation*}
%for all $(\kappa,\nu) \vdash n$
we have
\begin{equation*}
\iota_q^{\lambda,\mu}(U_{\alpha,\beta}) = \begin{cases}
1 &\text{if $(\lambda,\mu) = (\alpha,\beta)$},\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
%The matrix $(
Now define the $\mathbb Q(q)[\bn]$
%group algebra
\begin{equation*}
D = \sum_{(\alpha,\beta) \vdash n}
a_{\alpha,\beta}(q) U_{w_{\alpha,\beta}}.
%    g = \sum_{(\alpha,\beta) \vdash n}
%    \frac{a_{\alpha,\beta}(q)}{z_\alpha z_\beta 2^{\ell(\alpha) +\ell(\beta)}}
%    (U_{w_{\alpha,\beta}}).
\end{equation*}
%  ($\star$ No $q$ in the denominator?)
By (\ref{eq:iotadef}) and (\ref{eq:omegabc}) we have
\begin{equation*}
%  \begin{aligned}
Y_q(D) = \sum_{(\lambda,\mu) \vdash n}
\iota_q^{\lambda,\mu}
\Big( \sum_{(\alpha,\beta) \vdash n}
%\frac{a_{\alpha,\beta}(q)}{z_\alpha z_\beta 2^{\ell(\alpha) +\ell(\beta)}}
U_{\alpha,\beta} \Big) p_\lambda^+p_\mu^-
= \sum_{(\lambda,\mu) \vdash n} a_{\lambda,\mu}(q) p_\lambda^+p_\mu^-,
%    &= \sum_{(\lambda,\mu) \vdash n}
%    \frac{a_{\lambda,\mu}(q)\iota_q^{\lambda,\mu}(U_{\lambda,\mu})}
%         {z_\lambda z_\mu 2^{\ell(\lambda) +\ell(\mu)}}
%         p_\lambda^+(x,y)p_\mu^-(x,y)\\
%    = \sum_{(\lambda,\mu) \vdash n}
%\frac{a_{\lambda,\mu}}{z_\lambda z_\mu2^{\ell(\lambda) +\ell(\mu)}}
%    \iota^{\lambda,\mu}(w) = \begin{cases}
%    z_\lambda z_\mu 2^{\ell(\lambda)+\ell(\mu)} &\text{if $\sct(w) = (\lambda,\mu)$},\\
%    0 &\text{otherwise}.
%  \end{cases}
%    \end{aligned}
\end{equation*}
as desired.
\end{proof}

%($\star$ Explain the Stembridge-Remmel approach somewhere.)

%($\star$ Define chromatic symmetric functions of type $\msfBC$.  State that
%colorings correspond to evaluations of $(\epsilon\epsilon)^{\lambda,\mu}(\btc w1)$
%as a result or as a comment.)

For any type-$\msfBC$
incomparability graph $\Gamma = (V,E)$,
we define the
{\em type-$\msfBC$ chromatic symmetric function of $\Gamma$} to be
\begin{equation}\label{eq:XBCGdef}
%  X^{\msfBC}_G \defeq \sum_{\kappa, \kappa'} x_{\kappa(1)} \cdots x_{\kappa(n)},
X^{\msfBC}_G \defeq \sum_\kappa
(x_1^{|\kappa^{-1}(1)|} x_2^{|\kappa^{-1}(2)|} \cdots )
(y_1^{|\kappa^{-1}(-1)|} y_2^{|\kappa^{-1}(-2)|} \cdots ),
\end{equation}
where the sum is over all proper $\msfBC$-colorings
$\kappa: V \rightarrow \mathbb Z \ssm \{0 \}$ of $\Gamma$
(\S \ref{ss:bcincg}).
%Define a $\msfBC$-coloring $\kappa$ to be proper if
%\item $(a,b) \in E$ implies that $\kappa(a) \neq \kappa(b)$,
%\item for each grounded element $a \in V$ we have $\kappa(a) > 0$.

For the incomparability graph $\inc(Q)$ of a $\msfBC$-poset $Q$,
we may express the symmetric function
%, we may equivalently express
$X^{\msfBC}_{\inc(Q)}$ in terms of decompositions of $P$ into chains.
Letting $c_{Q,\lambda,\mu}$ be the number of column-strict marked $Q$-bitableaux
%(in which grounded elements appear only in left tableau?)
of shape $(\lambda^\tr, \mu^\tr)$, we have
%Then we have
\begin{equation*}
X^{\msfBC}_{\inc(Q)} = \sum_\lambda c_{Q,\lambda,\mu} (mm)_{\lambda,\mu}.
\end{equation*}

%($\star$ If we write the $\mu$ tableau backwards and to the left of the
%$\lambda$ tableau, does this make the inversion statistic come out correctly?)

\begin{thm}\label{t:bcxy}
For $w \in \bn$ \avoidingp{} and $Q = Q(w)$ defined as in
Subsection~\ref{ss:bcuio}, we have
$Y^{\msfBC}(\btc w1) = X^{\msfBC}_{\inc(Q)}$.
\end{thm}
\begin{proof}
The coefficient of $(mm)_{\lambda,\mu}$ in $X^{\msfBC}_{\inc(Q)}$
equals the number
of proper $\msfBC$-colorings of $\inc(Q)$ of type $(\lambda,\mu)$.
This is exactly the number of marked column-strict
$\inc(Q)$-bitableaux of shape $(\lambda^\tr,\mu^\tr)$.
By Theorem~\ref{t:epsiloneta},
this is $(\epsilon\epsilon)^{\lambda,\mu}(\btc w1)$,
and by (\ref{eq:YBCdefn}),
%the definition of $Y^{\msfBC}(\btc wq)$,
we have the desired equality.
\end{proof}
It would be interesting to state and prove
a $q$-analog of Theorem~\ref{t:bcxy}.
\begin{prob}\label{p:qanalogofbcchrom}
Define a statistic $\stat$ on proper $\msfBC$-colorings $\kappa$ of
type-$\msfBC$ interval graphs so that
for $w \in \bn$ \avoidingp,
$Q = Q(w)$ and
%for $P$ a $\msfBC$-unit interval order
%labeled as in (\ref{eq:altrespect}) and
$\Gamma = \inc(Q)$,
we have a $q$-chromatic symmetric function of the form
\begin{equation*}
X^{\msfBC}_{\Gamma,q} = \sum_\kappa q^{\stat(\kappa)}
(x_1^{|\kappa^{-1}(1)|} x_2^{|\kappa^{-1}(2)|} \cdots )
(y_1^{|\kappa^{-1}(-1)|} y_2^{|\kappa^{-1}(-2)|} \cdots )
\end{equation*}
%  where the sum is over all proper $\msfBC$-colorings
%  $\kappa: V \rightarrow \mathbb Z \ssm \{0 \}$ of $G$,
%  and
which satisfies the identity
%  \begin{equation*}
$Y_q^{\msfBC}(\btc wq) = X^{\msfBC}_{\Gamma,q}$.
%  \end{equation*}
\end{prob}

%($\star$ Include results, problems about generalizing properties
%of chromatic symmetric functions of type $\msfA$.)
%\item connection to a chromatic polynomial of type $\msfBC$
%\item plug in $-1$ to the above

\ssec{Another approach to type-$\msfBC$ symmetric functions}

From the plethystic power sum symmetric functions
%  \bigcup_{k \geq 0}
$  \{ p_\lambda^+ \,|\, \lambda \vdash k\}
\cup
%\bigcup_{k \geq 0}
\{ p_\lambda^- \,|\, \lambda \vdash k\}
\subseteq \Lambda_k(x,y)$,
one can define other plethystic
%Schur, homogeneous, etc.,
symmetric functions~\cite[\S 3]{RemTrans}, [62]
\begin{equation*}
\begin{gathered}
\{s_\lambda^+ \,|\, \lambda \vdash k\}, \quad
\{h_\lambda^+ \,|\, \lambda \vdash k\}, \quad
\{e_\lambda^+ \,|\, \lambda \vdash k\}, \quad
\{m_\lambda^+ \,|\, \lambda \vdash k\}, \quad
\{f_\lambda^+ \,|\, \lambda \vdash k\},\\ \quad
\{s_\lambda^- \,|\, \lambda \vdash k\}, \quad
\{h_\lambda^- \,|\, \lambda \vdash k\}, \quad
\{e_\lambda^- \,|\, \lambda \vdash k\}, \quad
\{m_\lambda^- \,|\, \lambda \vdash k\}, \quad
\{f_\lambda^- \,|\, \lambda \vdash k\}
\end{gathered}
\end{equation*}
to be those symmetric functions in $\Lambda_k(x,y)$ related to
$\{p_\lambda^+ \,|\, \lambda \vdash k \}$
$\{p_\lambda^- \,|\, \lambda \vdash k \}$
just as
$\{s_\lambda \,|\, \lambda \vdash k\}$,
$\{h_\lambda \,|\, \lambda \vdash k\}$,
$\{e_\lambda \,|\, \lambda \vdash k\}$,
$\{m_\lambda \,|\, \lambda \vdash k\}$,
$\{f_\lambda \,|\, \lambda \vdash k\}$
in $\Lambda_k(x)$ are related to
$\{p_\lambda \,|\, \lambda \vdash k \}$.
Such functions often appear in the literature as
$s_\lambda[X+Y], \dotsc, f_\lambda[X+Y]$
and $s_\lambda[X-Y],\dotsc,f_\lambda[X-Y]$.
Certain products of pairs of these
%plethystic symmetric functions
form nine more {\em plethystic} bases of the space $\Lambda_n(x,y)$:
\begin{equation}\label{eq:bcpbases}
\begin{gathered}
\{s_\lambda^+s_\mu^- \,|\, (\lambda,\mu) \vdash n \}\\
\{h_\lambda^+h_\mu^- \,|\, (\lambda,\mu) \vdash n \}, \quad
\{h_\lambda^+e_\mu^- \,|\, (\lambda,\mu) \vdash n \},\quad
\{e_\lambda^+h_\mu^- \,|\, (\lambda,\mu) \vdash n \},\quad
\{e_\lambda^+e_\mu^- \,|\, (\lambda,\mu) \vdash n \},\\
\{m_\lambda^+m_\mu^- \,|\, (\lambda,\mu) \vdash n \},\quad
\{m_\lambda^+f_\mu^- \,|\, (\lambda,\mu) \vdash n \},\quad
\{f_\lambda^+m_\mu^- \,|\, (\lambda,\mu) \vdash n \},\quad
\{f_\lambda^+f_\mu^- \,|\, (\lambda,\mu) \vdash n \}.
\end{gathered}
\end{equation}
Naturally, one may study $\Lambda_n(x,y)$ in terms of these bases
\begin{equation*}
\{ p_\lambda^+ p_\mu^- \,|\, (\lambda,\mu) \vdash n\}, \quad
\{ (pp)_{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n\},
\end{equation*}
instead of using the eleven bases in (\ref{eq:bcnpbases}) -- (\ref{eq:pbasis}).
It is straightforward to show
that matrices relating bases of the forms
$\{ o_\lambda^+g_\mu^- \,|\, (\lambda,\mu) \vdash n \}$
$\{ (pp)_{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$
%in (\ref{eq:ninemorebases})
are the same as those relating bases of the forms
$\{ (og)_{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$
%$\{ 2^{-\ell(\lambda)-\ell(\mu)} p_\lambda^+p_\mu^- \,|\, (\lambda,\mu) \vdash n \}$.
$\{ p_\lambda^+p_\mu^-/2^{\ell(\lambda)+\ell(\mu)} \,|\, (\lambda,\mu) \vdash n \}$.
Formulas for the matrix entries are given in \cite[App.\,A]{RemTrans}.

A correspondence between the plethystic bases and those of the trace space
$\trsp(\hbnq)$ is given explicitly by the ($q$-extension of the)
{\em nonplethystic} $\msfBC$-Frobenius map
\cite[\S 3]{RemTrans}, [62]
%($\star$ cite Remmel and Stembridge)
\begin{align}
\npfrobch_q\ntksp:
\trsp(\hbnq) &\rightarrow \Lambda_n(x,y) \label{eq:afrobchaaer0}\\
\npfrobch_q(\theta_q) &= \frac1{2^nn!}
\ntnsp \sum_{w \in \bn} \ntksp 2^{\ell(\alpha(w)) + \ell(\beta(w))} \theta(w)
(pp)_{\alpha(w),\beta(w)} \label{eq:afrobchaaer1} \\
&= \nTksp \sum_{(\lambda,\mu) \vdash n} \ntnsp
\frac{\theta(\lambda,\mu)}{z_\lambda z_\mu}
(pp)_{\lambda,\mu}, \label{eq:afrobchaaer2}
\end{align}
analogous to (\ref{eq:frobchaaer0}) -- (\ref{eq:frobchaaer2}).
%where $\alpha(w)$, $\beta(w)$ are the partitions satisfying
%$\sct(w) = (\alpha(w), \beta(w))$, and where we define
%$\theta(\lambda,\mu) \defeq \theta(w)$ for any $w \in \bn$
%having $\sct(w) = (\lambda,\mu)$.
Specifically, $\npfrobch_q$ maps
\begin{equation*}
\begin{gathered}
\quad (\epsilon\epsilon)^{\lambda,\mu}_q \mapsto e_\lambda^+e_\mu^-,
\quad (\epsilon\eta)^{\lambda,\mu}_q \mapsto e_\lambda^+h_\mu^-,
\quad (\eta\epsilon)^{\lambda,\mu}_q \mapsto h_\lambda^+e_\mu^-,
\quad (\eta\eta)^{\lambda,\mu}_q \mapsto h_\lambda^+h_\mu^-,\\
(\phi\phi)^{\lambda,\mu}_q \mapsto m_\lambda^+m_\mu^-,
\quad (\phi\gamma)^{\lambda,\mu}_q \mapsto m_\lambda^+f_\mu^-,
\quad (\gamma\phi)^{\lambda,\mu}_q \mapsto f_\lambda^+m_\mu^-,%\\
\quad (\gamma\gamma)^{\lambda,\mu}_q \mapsto f_\lambda^+f_\mu^-,\\
%  \dotsc,
(\chi\chi)^{\lambda,\mu}_q \mapsto s_\lambda^+s_\mu^-,
\quad (\psi\psi)^{\lambda,\mu}_q \mapsto p_\lambda^+p_\mu^-,
\quad \iota^{\lambda,\mu}_q \mapsto 2^{\ell(\lambda)+\ell(\mu)}(pp)_{\lambda,\mu}.
\end{gathered}
\end{equation*}
Defining a trace generating function in $\zqq \otimes \Lambda_n(x,y)$
%($\star$ Think about coefficient ring.)
in terms of characters $(\epsilon\epsilon)_q^{\lambda,\mu}$, plethystic monomial
symmetric functions $m_\lambda^+m_\mu^-$, and elements $D \in \hbnq$, we have
expansions analogous to those in Proposition~\ref{p:YBCexpansions},
%($\star$ Check this.)
\begin{equation*}
\begin{aligned}
&\sum_{(\lambda,\mu)\vdash n} (\epsilon\epsilon)_q^{\lambda,\mu}(D) m_\lambda^+m_\mu^-
= \sum_{(\lambda,\mu)\vdash n} (\epsilon\eta)_q^{\lambda,\mu}(D) m_\lambda^+f_\mu^-
%  = \sum_{(\lambda,\mu)\vdash n} (\eta\epsilon)_q^{\lambda,\mu}(D) f_\lambda^+m_\mu^-\\
%  &= \sum_{(\lambda,\mu)\vdash n} (\eta\eta)_q^{\lambda,\mu}(D) f_\lambda^+f_\mu^-
%  = \sum_{(\lambda,\mu)\vdash n} (\chi\chi)_q^{\lambda^\tr\ntnsp,\mu^\tr}(D) s_\lambda^+s_\mu^-
%  = \sum_{(\lambda,\mu)\vdash n} (\phi\phi)_q^{\lambda,\mu^-}(D) e_\lambda^+e_\mu^-\\
%  &= \sum_{(\lambda,\mu)\vdash n} (\phi\gamma)_q^{\lambda,\mu}(D) e_\lambda^+h_\mu^-
%  = \sum_{(\lambda,\mu)\vdash n} (\gamma\phi)_q^{\lambda,\mu}(D) h_\lambda^+e_\mu^-
= \cdots
= \sum_{(\lambda,\mu)\vdash n} (\gamma\gamma)_q^{\lambda,\mu}(D) h_\lambda^+h_\mu^-\\
&= \sum_{(\lambda,\mu)\vdash n}
\frac{(-1)^{\ell(\lambda)+\ell(\mu)}(\psi\psi)_q^{\lambda,\mu}(D)}{z_\lambda z_\mu}
p^+_\lambda p^-_\mu
= \sum_{(\lambda,\mu)\vdash n}
(-1)^{\ell(\lambda)+\ell(\mu)} \iota_q^{\lambda,\mu}(D)
\end{aligned}
\end{equation*}
%\section{Chromatic symmetric functions of type $BC$}
%($\star$ Just incorporate this into previous section?)
%Would the notation $a_{\lambda}^+b_\mu^-$ make sense?)
%($\star$ Make some concluding remark, such as that
While one could define $Y^{\msfBC}_q(D)$ to be the above symmetric function
%this way
%as in (\ref{eq:YBCdefnalt})
instead of
that in (\ref{eq:YBCdefn}), this would lead to a less natural
connection to the
%in terms of the plethystic monomial basis.  The
$\msfBC$-chromatic symmetric
function (\ref{eq:XBCGdef}).
%would be less natural this way.  Are the quasi-symmetric functions
%analogous to these?)

\section{Hessenberg varieties}\label{s:hess}

%($\star$ Rewrite this with better notation: right now there are too many H's?)

%($\star$ Improve this.)
Hessenberg varieties are subvarieties of flag varieties which were first
studied~[18], [19]
in conjunction with questions
concerning eigenvalues of linear operators.
%problems in
%numerical analysis.
Later work revealed connections to other varieties and representation theory.
%and chromatic symmetric functions.
%Eventually it was discovered that they are special
%cases of Lusztig varieties ($\star$ cite where these varieties were defined?)
%and that they are related to type-$\msfA$
%chromatic symmetric functions.
Hessenberg varieties of Coxeter type $\msfA$, $\msfBB$, $\sfC$
are parametrized by appropriate
words $m = m_h \cdots m_l$
called {\em Hessenberg words}.  More precisely, given
%($\star$ need modifiers like regular semisimple?)
Lie group $\mathrm G$
%\in \{ \mathrm{GL}_n(\mathbb C),
%\mathrm{SO}_{2n+1}(\mathbb C),
%\mathrm{SP}_n(\mathbb C) \}$
and Borel subgroup $\mathrm B$ of upper triangular matrices in $\mathrm G$
%Lie algebras
(in the sense of \cite[\S 21.2]{Hum-LAG}),
let $\mathfrak g$, $\mathfrak b$
be the Lie algebras of these,
%and let $W = \sn$ or $\bn$ be the Weyl group of $\mathrm G$.
and for each
word $m$
%= m_h \cdots m_l$
define the subspace
\begin{equation}\label{eq:hspace}
\HH_m \defeq \{ Z = (z_{i,j}) \in \mathfrak g \,|\, z_{i,j} = 0 \text{ for all }
i > m_i \}
\end{equation}
of $\mathfrak g$. Call this a {\em Hessenberg space} if it satisfies
%Precise descriptions of which words $m$ are valid Hessenberg words follow from
the Lie algebra containment conditions
\begin{equation}\label{eq:contcond}
\mathfrak b \subseteq \HH_m, \qquad
\big[ \mathfrak b, \HH_m \big] \subseteq \HH_m.
\end{equation}
These conditions determine which words $m$ are valid Hessenberg words.
%imply conditions
%which imply other conditions on the components of $m$.
For an appropriate
%$n \times n$
complex diagonal matrix $S$ and Hessenberg word $m$,
a subvariety $\hess(m) = \hess(S,m)$ of the flag variety
%$\mathcal F = \mathrm G/\mathrm B$ is defined by
$\mathrm G/\mathrm B$ is defined by
%Specifically, we have
\begin{equation}\label{eq:hessdef}
\hess(m) \defeq \{ g\mathrm B \in
%  \mathcal F
\mathrm G/\mathrm B
\,|\, g \in \mathrm G,\ g^{-1}\ntnsp Sg \in \HH_m \},
\end{equation}
and is called the {\em regular semisimple Hessenberg variety associated to $m$}.
%($\star$ Cite Lakshimbai for notation/definitions?)
%($\star$ Cite some paper on Hessenberg varieties?)

The cohomology of $\hess(m)$ vanishes in odd degree,
%Specifically, for $w$, $m$ related as in (\ref{eq:wtom}), we have
\begin{equation}\label{eq:cohvanish}
\coh^*(\mathcal \hess(m)) = \bigoplus_{j\geq0} \coh^{2j}(\hess(m)),
\end{equation}
and for $W$ the Weyl group of $G$,
each graded component has the structure of a $W$-module
%n $\sn$-module
via an action defined by Tymoczko~[66], [67].
%($\star$ Was the action defined by
%Balibanu and Crooks~\cite{BCPerverse}?)
\begin{equation}\label{eq:frobcharhess}
\mathrm{ch}(\coh^{2j}(\hess(m)))
\end{equation}
be the Frobenius characteristic of the character
of the module $\coh^{2j}(\hess(m))$.
For Hessenberg varieties of type $\msfA$,
the grading (\ref{eq:cohvanish}) is
%known to be
closely connected to trace evaluations and graph coloring.
For Hessenberg varieties of types $\msfBB$ and $\msfC$,
no such connection is known.

\ssec{Hessenberg varieties of type $\msfA$}

Define type-$\msfA$ Hessenberg varieties,
%we follow
as in
(\ref{eq:hspace}), (\ref{eq:hessdef}) with
$\mathrm G = \mathrm{GL}_n(\mathbb C)$,
$\mathfrak g = \mathfrak{gl}_n(\mathbb C)$,
%$W = \sn$,
and $S = \diag(\lambda_1,\dotsc,\lambda_n)$,
%where $\lambda_1,\dotsc,\lambda_n \in \mathbb C$ are distinct.
where $\lambda_1,\dotsc,\lambda_n$ are distinct.
%$ any $n \times n$ diagonal matrix having distinct
These choices and (\ref{eq:contcond}) imply
type-$\msfA$ Hessenberg words to be those of the form
%Call a word
$m = m_1 \cdots m_n \in \mathbb [n]^n$
%a (type-$\msfA$)
%{\em Hessenberg word} if it
%the conditions
\begin{enumerate}
\item $m_i \geq i$ for $i = 1,\dotsc,n$,
\item $m_1 \leq \cdots \leq m_n = n$,
\end{enumerate}
%These words
%Hessenberg words
and corresponding bijectively to $312$-avoiding permutations
%$w_1 \cdots w_n \in \sn$
$w \in \sn$
by the map
\begin{equation}\label{eq:wtom}
m_i = \max \{ w_1,\dotsc, w_i \},
\end{equation}
as in Algorithm~\ref{a:wtop}.
%(See (\ref{eq:firsthess}).)
Write $\ahess(m)$ for the regular semisimple type-$\msfA$ Hessenberg
variety associated to $m$.

%To each Hessenberg word $m$ we associate
%a subspace of $\mathfrak{gl}_n(\mathbb C)$
%called the {\em Hessenberg space indexed by $m$},
%  \HH_m \defeq \{ Z = (z_{i,j}) \in \mathfrak{gl}_n(\mathbb C) \,|\, z_{i,j} = 0 \text{ for all } i > m_j \}.
%Letting $\mathrm B$ be the Borel subgroup of $\mathrm{GL}_n(\mathbb C)$
%consisting of upper triangular matrices,
%we also
%associate to $m$ a subvariety of the flag variety
%$\mathcal F_n = \mathrm{GL}_n(\mathbb C)/\mathrm B$.
%Fix any diagonal matrix
%$S \in \mathrm{GL}_n(\mathbb C)$ having distinct eigenvalues and define the
%{\em regular semisimple type-$\msfA$ Hessenberg variety associated to $m$}
%to be the set of cosets
%  \hess(m) = \hess(S,m) \defeq
%  \{ g\mathrm B \in \mathcal F_n  \,|\, g \in \mathrm{GL}_n(\mathbb C),\,
%  g^{-1}Sg \in \HH_m \}.
%($\star$ See Lakshmibai and mention Lie algebra
%if necessary.  Mention other papers about
%Hessenberg varieties?)
%($\star$ Omit this? letting $\mathfrak b$
%be its Lie algebra ($\star$ add precise description? or citation?),
%we have that conditions (1) -- (2) above are equivalent to the containments
%  \mathfrak b \subseteq \HH_m, \qquad [\mathfrak b, \HH_m] \subseteq \HH_m.
%  \end{equation*}

%The cohomology of $\hess(m)$ vanishes in odd degree.
%Specifically, for $w$, $m$ related as in (\ref{eq:wtom}), we have
%  \coh^*(\mathcal \hess(m)) = \bigoplus_{j=0}^{\ell(w)} \coh^{2j}(\hess(m)),
%with each graded component having the structure of an $\sn$-module
%via an action defined by Tymoczko~[66], [67].
%Let $\mathrm{ch}(\coh^{2j}(\hess(m)))$ be
%the Frobenius characteristic of the character
%of the module $\coh^{2j}(\hess(m))$.
%($\star$ Explain that the notation
Shareshian and Wachs conjectured~\cite[Conj.\,10.1]{SWachsChromQF}
that for $w \in \sn$ corresponding to $m$,
the generating function
\begin{equation}\label{eq:gradedchA}
\mathrm{Fr}^{\msfA}(m;q) \defeq
\sum_{j=0}^{\ell(w)} \mathrm{ch}(\coh^{2j}(\ahess(m)))q^j
%  \in \Lambda_n \otimes \mathbb Z[q]
\end{equation}
for the type-$\msfA$ Frobenius characteristics
of the $\sn$-dot action
%in type $\msfA$
is equal to a chromatic symmetric function,
%$\omega X_{\inc(P),q}$,
%\in \mathbb Z[q] \otimes \Lambda_n$
%\in \Lambda_n \otimes \mathbb Z[q] \cong \Lambda_n[q]$
viewed as a polynomial in $q$ with coefficients
%are symmetric functions
in $\Lambda_n$.
%, we obtain
This was proved by Brosnan--Chow~[13]
and Guay-Paquet~[28].
Combining the equality with Theorem~\ref{t:YequalsX}
we specifically have the following.
%Cite this theorem appropriately.  It was conjectured by Shareshian-Wachs
%in \cite[Conj.\,10.1]{SWachsChromQF}.
\begin{thm}\label{t:tymXY}
Fix $w \in \sn$ avoiding the pattern $312$,
let $P$ be its corresponding unit interval order,
and define $m$ as in (\ref{eq:wtom}).
%  and let $e$ be the number of edges in $\inc(P)$.
We have
\begin{equation}\label{eq:tymXY}
\mathrm{Fr}^{\msfA}(m;q)
%    \sum_{j=0}^{\ell(w)} \ntksp \mathrm{ch}(\coh^{2j}(\hess^{\mathsf{A}}(m)))q^j
= \omega X_{\inc(P),q}
= \omega Y_q(\wtc wq).
\end{equation}
\end{thm}
We can see in three ways that the function in (\ref{eq:tymXY})
belongs to $\spn_{\mathbb N[q]} \{ s_\lambda \,|\, \lambda \vdash n \}$:
\begin{enumerate}
\item The character of
%any $\sn$-module such as
belongs to $\spn_{\mathbb N}\{ \chi^\lambda \,|\, \lambda \vdash n \}$.
Thus its Frobenius characteristic (\ref{eq:frobchs})
%the sum is a polynomial in $q$ with coefficients in
belongs to
$\spn_{\mathbb N}\{ s_\lambda \,|\, \lambda \vdash n \}$.
%  with Schur-positive coefficients.
\item By \cite[Thm.\,6.3]{SWachsChromQF} the coefficient of
$s_\lambda$ in $X_{\inc(P),q}$ belongs to $\mathbb N[q]$.
\item By \cite[Lem.\,1.1]{HaimanHecke} we have
$\chi_q^\lambda(\wtc wq) \in \mathbb N[q]$.
\end{enumerate}
Furthermore, the function (\ref{eq:tymXY}) is conjectured
%(\cite[Conj.\,5.1]{SWachsChromQF} and elsewhere)
to belong to $\spn_{\mathbb N[q]} \{ h_\lambda \,|\, \lambda \vdash n \}$.
%(See, e.g.,
%(See, e.g., $\star$ Stanley, $\star$ Stembridge, \cite
%One might try to prove this as follows.
This open problem also can be viewed in three ways.
\begin{prob}
Prove one of the following.
\begin{enumerate}
\item \cite[Conj.\,10.4]{SWachsChromQF}
For each
%Hessenberg function $m$,
%its corresponding
permutation $w \in \sn$ avoiding the pattern $312$,
Hessenberg word $m$ defined as in (\ref{eq:wtom}),
and $j = 0,\dotsc,\ell(w)$,
the $\sn$-module $\coh^{2j}(\hess^{\msfA}(m))$ is a permutation module in which
each point stabilizer is a Young subgroup.
\item %Show that
\cite[Conj.\,5.1]{SWachsChromQF}
For each unit interval order $P$ labeled as in
the function $X_{\inc(P),q}$ belongs to
$\spn_{\mathbb N[q]} \{ e_\lambda \,|\, \lambda \vdash n \}$.
\item %Show that
\cite[Conj.\,2.1]{HaimanHecke}
For each permutation $w \in \sn$ and each partition $\lambda \vdash n$,
we have $\phi_q^\lambda(\wtc wq) \in \mathbb N[q]$.
\end{enumerate}
\end{prob}
The statements above satisfy the implications
$(1) \Leftrightarrow (2) \Leftarrow (3)$.
%($\star$ right?)
%($\star$ Should the first implication be an equivalence?)
For progress on these problems, see, e.g., [2],
[30], [51], and references
cited in \cite[\S 3.5]{SkanCCS}.
%($\star$ Cite some paper. By Martha or Abreu-Nigro
%($\star$ and others?)

\ssec{Hessenberg varieties of types $\msfBB$ and $\msfC$}

%($\star$ Fix this for type $\msfBB$.)
Define type-$\msfBB$ Hessenberg varieties
%we follow
as in
(\ref{eq:hspace}), (\ref{eq:hessdef}) with
$\mathrm G = \mathrm{SO}_{2n+1}(\mathbb C)$,
$\mathfrak g = \mathfrak{so}_{2n+1}(\mathbb C)$,
%$W = \bn$,
$S = \diag(0, \lambda_1, \dotsc, \lambda_n, -\lambda_1, \dotsc, -\lambda_n)$,
where $\lambda_1,\dotsc,\lambda_n$ are chosen so that the $2n+1$
%and $0, \lambda_1, \dotsc, \lambda_n, -\lambda_1, \dotsc, -\lambda_n$ distinct.
%where these
diagonal entries are distinct.
%$\lambda_1,\dotsc,\lambda_n \in \mathbb C \ssm \{0\}$ satisfy $\lambda_i \neq \pm \lambda_j$ for $i \neq j$.
%$ any $(2n+1) \times (2n+1)$
%diagonal matrix having distinct
%eigenvalues $
%the first $n$ of which have distinct absolute values.
%($\star$ Is this right?).
%($\star$ This isn't precise.
%What needs to be said about eigenvalue repetition and the one
%eigenvalue which is $0$.  See Erdmann and Wildon.)
These choices and (\ref{eq:contcond}) imply
type-$\msfBB$ Hessenberg words to be those of the form
%Call a word
$m = m_{\ol n} \cdots m_{\ol1} m_0 m_1 \cdots m_n \in \mathbb
(\{0\} \cup [\ol n, n])^{2n+1}$
%  \end{equation*}
%a (type-$\msfA$)
%{\em Hessenberg word} if it
%the conditions
\begin{enumerate}
\item $m_i \geq i$ for $i = \ol n,\dotsc,\ol1,0,1,\dotsc,n$,
\item $m_{\ol n} \leq \cdots \leq m_{\ol1} \leq m_0 \leq m_1 \leq \cdots \leq m_n = n$,
\item the integer partition
$(m_n+n+1,\dotsc,m_1+n+1,m_0+n+1,m_{\ol1}+n+1, \dotsc, m_{\ol n}+n+1)$
%+ (n+1,\dotsc,n+1)$
is self-transpose.
%= (m_n,\dotsc,m_1,m_0,m_{\ol1}, \dotsc, m_{\ol n})$.
%  $m_i = \# \{ j \,|\, m_i \geq \ol j \}$ for $i \in \{0\} \cup [\ol n, n]$.
%($\star$ Fix this.)
\end{enumerate}
%It follows that for fixed $n$, the nuber of type-$\msfBB$ Hessenberg words
%above equals the number of symmetric Dyck paths from $(0,0)$ to $(2n+1,2n+1)$,
%i.e., $\tbinom{2n+1}n$.
Define type-$\msfC$ Hessenberg varieties
as in
(\ref{eq:hspace}), (\ref{eq:hessdef}) with
$\mathrm G = \mathrm{SP}_{2n}(\mathbb C)$,
$\mathfrak g = \mathfrak{sp}_{2n}(\mathbb C)$,
%$W = \bn$,
$S = \diag(\lambda_1, \dotsc, \lambda_n, -\lambda_1, \dotsc, -\lambda_n)$,
where $\lambda_1,\dotsc,\lambda_n$ are chosen so that the $2n$
diagonal entries are distinct.
%where $\lambda_1,\dotsc,\lambda_n \in \mathbb C \ssm \{0\}$
%satisfy $\lambda_i \neq \pm \lambda_j$ for $i \neq j$.
%$S$ any $2n \times 2n$ diagonal matrix having distinct nonzero eigenvalues,
%the first $n$ of which have distinct absolute values.
%($\star$ This isn't precise.
%What needs to be said about eigenvalue repetition?)
These choices and (\ref{eq:contcond}) imply
type-$\msfC$ Hessenberg words to be those of the form
%Call a word
$m = m_{\ol n} \cdots m_{\ol1}m_1 \cdots m_n \in \mathbb
[\ol n, n]^{2n}$
%  \end{equation*}
%a (type-$\msfA$)
%{\em Hessenberg word} if it
%the conditions
\begin{enumerate}
\item $m_i \geq i$ for $i = \ol n,\dotsc,n$,
\item $m_{\ol n} \leq m_{\ol1} \leq \cdots \leq m_1 \leq \cdots \leq m_n = n$,
\item the integer partition $(|[\ol n, m_n]|,\dotsc,|[\ol n, m_1]|,
|[\ol n, m_{\ol1}]|, \dotsc, |[\ol n, m_{\ol n}]|)$
%  $(m_n+n,\dotsc,m_1+n,m_{\ol 1}+n+1,\dotsc, m_{\ol n}+n+1)$
is self-transpose.
%  $m_i = \# \{ j \,|\, m_j > 10 - i \}$ for $i \in [\ol n, n]$.
%($\star$ Fix this.)
\end{enumerate}
Let $\mathcal M^{\msfBB}$,
$\mathcal M^{\msfC}$
denote the sets of type-$\msfBB$ and $\msfC$ Hessenberg words described above,
\begin{prop}
We have $|\mathcal M^{\msfBB}| = \tbinom{2n+1}n$, $|\mathcal M^{\msfC}| = \tbinom{2n}n$.
\end{prop}
\begin{proof}
Placing Young diagrams of the above self-transpose partitions
into Young diagrams of size $(2n+1) \times (2n+1)$
%, for type $\msfBB$,
$2n \times 2n$
%, for type $\msfC$,
defines a bijection with
palindromic Dyck paths of
length $4n+2$ or $4n$, respectively.
The numbers of these are $\tbinom{2n+1}n$ and $\tbinom{2n}n$
\end{proof}
%($\star$ Should the above partition be
%$m_{\ol n}+n,\dotsc,m_{\ol 1}+n, m_1+n,\dotsc,m_n+n$?)
\noindent
%we have a bijective correspondence between
type-$\msfA$ Hessenberg words correspond bijectively to
type-$\msfA$ codominant elements of $\sn$,
this bijection
does not extend
to types $\msfBB$ and $\msfC$:
%We may compute
%$|\mathcal M^{\msfBB}|$ and
%$|\mathcal M^{\msfC}|$
%In particular, one may count
%type-$\msfBB$ and type-$\msfC$ Hessenberg words
by Theorem~\ref{t:catalan}
%the number of $\msfBC$-codominant
%elements of $\bn$ is $\tfrac{1}{n+2}\tbinom{2n+2}{n+1}$,
the number of elements
%$w_{\ol n} \cdots w_{\ol1} w_1 \cdots w_n$ of
$w \in \bn$
\avoidingsignedp{} is $\tfrac{1}{n+2}\tbinom{2n+2}{n+1}$.

Write $\bhess(m)$
and $\chess(m)$
for the regular semisimple type-$\msfBB$
varieties associated to appropriate Hessenberg words $m$,
and define the generating functions
\begin{equation}\label{eq:gradedchBC}
\mathrm{Fr}^{\msfBB}(m;q)
\defeq
\sum_j \ntnsp \mathrm{ch}(\coh^{2j}(\bhess(m)))q^j,
%  = \omega Y_q(D_w).
\quad \
\mathrm{Fr}^{\sfC}(m;q)
\defeq
\sum_j \ntnsp \mathrm{ch}(\coh^{2j}(\chess(m)))q^j.
\end{equation}

\vspace{-1.5mm}
\noindent
(See \cite[\S 8]{PSPerverseNHV} for some computations.)
It would be interesting
to connect the symmetric functions
$\mathrm{Fr}^{\msfBB}(m;q)$, $\mathrm{Fr}^{\msfC}(m;q)$
to type-$\msfBC$ chromatic symmetric functions and
type-$\msfBC$ trace generating functions,
i.e., to formulate type-$\msfBB$ and $\msfC$ analogs
of Theorem~\ref{t:tymXY}.
By Theorem~\ref{t:bcxy}
the $q=1$ specializations of the latter two are equal;
general equality is stated as
%and the problem of proving equality in general is stated in
One could also consider
$\mathrm{Fr}^{\msfC}(m;q)$ as
trace generating functions.
By Proposition~\ref{p:everysymmfnbcq},
such interpretations must exist.
%we may ask for interpretations of
\bp\label{p:hesstrace}
%For some subsets
Find families
$\{ D^{\msfBB}_m \,|\, m \in \mathcal M^{\msfBB} \}$,
$\{ D^{\msfC}_m \,|\, m \in \mathcal M^{\msfC} \}$ of
elements of
whose trace generating functions satisfy
%such that we have
\begin{enumerate}
\item $\omega \YBCq(D^{\msfBB}_m) = \mathrm{Fr}^{\msfBB}(m;q)$ for all
$m \in \mathcal M^{\msfBB}$,
\item $\omega \YBCq(D^{\msfC}_m) = \mathrm{Fr}^{\msfC}(m;q)$ for all
$m \in \mathcal M^{\msfC}$.
\end{enumerate}
%  \qquad
%  \omega \YBCq(D_m) = \mathrm{Fr}^{\msfC}(m;q)
%  ~\forall m \in \mathcal M^{\msfC}.
%  \end{equation*}
\ep
%It would also be interesting to
\noindent
Third, one could try to interpret
%ask for interpretations of
%The problem of interpreting
$\mathrm{Fr}^{\msfBB}(m;q)$, $\mathrm{Fr}^{\sfC}(m;q)$
in terms of graph coloring~[52].

\bp\label{p:hesschrom}
Define families
$\mathcal G^{\msfBB} =\{ G^{\msfBB}(m) \,|\, m \in \mathcal M^{\msfBB} \}$,
$\mathcal G^{\msfC} =\{ G^{\msfC}(m) \,|\, m \in \mathcal M^{\msfC} \}$
of graphs and families
$\{ X_{G,q}^{\msfBB} \,|\, G \in \mathcal G^{\msfBB} \}$,
$\{ X_{G,q}^{\msfC} \,|\, G \in \mathcal G^{\msfC} \}$
of chromatic symmetric functions
so that we have
\begin{enumerate}
\item $\omega X^{\msfBB}_{G^{\msfBB}(m),q} = \mathrm{Fr}^{\msfBB}(m;q)$ for all
$m \in \mathcal M^{\msfBB}$,
\item $\omega X^{\msfC}_{G^{\msfC}(m),q} = \mathrm{Fr}^{\msfC}(m;q)$ for all
$m \in \mathcal M^{\msfC}$.
\end{enumerate}
\ep
\noindent
%In addition,
it would also be interesting to connect these problems to
the Kazhdan--Lusztig basis of $\hbnq$ and to the type-$\msfBC$
indifference graphs defined in Subsection~\ref{ss:bcincg}.
\begin{prob}
To what extent can the symmetric functions in
%Decide to what extent the definition () of $X^{\msfBC}_{G,q}$ solves
Problems~\ref{p:hesstrace} -- \ref{p:hesschrom}
be chosen to simultaneously solve Problem~\ref{p:qanalogofbcchrom}?
%that their $q=1$ specializations have the form
%$X^{\msfBC}_G$ defined in (\ref{eq:XBCGdef})?
\end{prob}

%\text{ a type-} \mathsf{B} \text{ Hessenberg word } \}$
%  of graphs and a family of chromatic symmetric functions

%see that for fixed $n$,
%the numbers of

%from $(0,0)$ to $(0,4n+2)$,

\section{Acknowledgements}

The authors are grateful to
Jonathan Boretsky,
Patrick Brosnan,
Mahir Can,
Tim Chow,
Theodossios Douvropoulos,
Emma Gray,
Angela Hicks,
Jongwon Kim,
Diep Luong,
Jeremy Martin,
Nicholas Mayers,
Antonio Nigro,
Rosa Orellana,
Tommy Parisi,
Martha Precup,
Arun Ram,
Bruce Sagan,
George Seelinger,
John Shareshian,
Eric Sommers,
John Stembridge,
Michelle Wachs,
Jiayuan Wang,
Andy Wilson
for helpful conversations.

\begin{thebibliography}{10}

[1]
{\sc A.~Abreu and A.~Nigro}.
\newblock An update on {Haiman's} conjectures (2022).
\newblock Preprint {\tt math.AG/2206.00073} on {ArXiv}.

[2]
{\sc A.~Abreu and A.~Nigro}.
\newblock Splitting the cohomology of {Hessenberg} varieties and $e$-positivity
of chromatic symmetric functions (2023).
\newblock Preprint {\tt math.AG/2304.10644} on {ArXiv}.

[3]
{\sc R.~M. Adin, C.~A. Athanasiadis, S.~Elizalde, and Y.~Roichman}.
\newblock Character formulas and descents for the hyperoctahedral group.
\newblock {\em Adv. in Appl. Math.\/}, {\bf 87} (2017) pp. 128--169.

[4]
{\sc D.~Anderson and W.~Fulton}.
\newblock Degeneracy loci, pfaffians, and vexillary signed permutations (2012).
\newblock Preprint {\tt math.AG:1210.2066} on {ArXiv}.

[5]
{\sc D.~Anderson and W.~Fulton}.
\newblock Vexillary signed permutations revisited.
\newblock {\em Algebr. Comb.\/}, {\bf 3}, 5 (2020) pp. 1041--1057.

[6]
{\sc C.~A. Athanasiadis}.
\newblock Power sum expansion of chromatic quasisymmetric functions.
\newblock {\em Electron. J. Combin.\/}, {\bf 22}, 2 (2015).
\newblock Paper 2.7, 9 pages.

[7]
{\sc D.~Beck, J.~Remmel, and T.~Whitehead}.
\newblock The combinatorics of transition matrices between the bases of the
symmetric functions and the {$B_n$} analogues.
\newblock {\em Discrete Math.\/}, {\bf 153} (1996) pp. 3--27.

[8]
{\sc S.~Billey and V.~Lakshmibai}.
\newblock {\em Singular loci of {S}chubert varieties\/}, vol. 182 of {\em
Progress in Mathematics\/}.
\newblock Birkh\"auser Boston Inc., Boston, MA (2000).

[9]
{\sc S.~Billey and T.~K. Lam}.
\newblock Vexillary elements in the hyperoctahedral group.
\newblock {\em J. Algebraic Combin.\/}, {\bf 8}, 2 (1998) pp. 139--152.

[10]
{\sc S.~C. Billey}.
\newblock Pattern avoidance and rational smoothness of {Schubert} varieties.
\newblock {\em Adv. Math.\/}, {\bf 139} (1998) pp. 141--156.

[11]
{\sc S.~C. Billey and G.~Warrington}.
\newblock {Kazhdan-Lusztig} polynomials for $321$-hexagon-avoiding
\newblock {\em J. Algebraic Combin.\/}, {\bf 13}, 2 (2001) pp. 111--136.

[12]
{\sc A.~Bj{\"o}rner and F.~Brenti}.
\newblock {\em Combinatorics of {C}oxeter groups\/}, vol. 231 of {\em Graduate
Texts in Mathmatics\/}.
\newblock Springer, New York (2005).

[13]
{\sc P.~Brosnan and T.~Y. Chow}.
\newblock Unit interval orders and the dot action on the cohomology of regular
semisimple {H}essenberg varieties.
\newblock {\em Adv. Math.\/}, {\bf 329} (2018) pp. 955--1001.

[14]
{\sc S.~Clearman, M.~Hyatt, B.~Shelton, and M.~Skandera}.
\newblock Evaluations of {H}ecke algebra traces at {K}azhdan-{L}usztig basis
\newblock {\em Electron. J. Combin.\/}, {\bf 23}, 2 (2016).
\newblock Paper 2.7, 56 pages.

[15]
{\sc A.~Clearwater and M.~Skandera}.
\newblock Total nonnegativity and {H}ecke algebra trace evaluations.
\newblock {\em Ann.\ Combin.\/}, {\bf 25} (2021) pp. 757--787.

[16]
{\sc V.~E. Coll, Jr., N.~W. Mayers, and N.~Russoniello}.
\newblock The index and spectrum of {L}ie poset algebras of types {B}, {C}, and
\newblock {\em Electron. J. Combin.\/}, {\bf 28}, 3 (2021).
\newblock Paper 3.47, 23 pages.

[17]
{\sc S.~Csar}.
\newblock {\em Root and weight semigroup rings for signed posets\/}.
\newblock Ph.D. thesis, University of Minnesota, Minneapolis, MN (2014).

[18]
{\sc F.~De~Mari, C.~Procesi, and M.~A. Shayman}.
\newblock Hessenberg varieties.
\newblock {\em Trans. Amer. Math. Soc.\/}, {\bf 332}, 2 (1992) pp. 529--534.

[19]
{\sc F.~De~Mari and M.~A. Shayman}.
\newblock Generalized {E}ulerian numbers and the topology of the {H}essenberg
variety of a matrix.
\newblock {\em Acta Appl. Math.\/}, {\bf 12}, 3 (1988) pp. 213--235.

[20]
{\sc V.~Deodhar}.
\newblock A combinatorial setting for questions in {Kazhdan--Lusztig} theory.
\newblock {\em Geom. Dedicata\/}, {\bf 36}, 1 (1990) pp. 95--119.

[21]
{\sc R.~Dipper and G.~James}.
\newblock Representations of {H}ecke algebras of type {$B_n$}.
\newblock {\em J.\ Algebra\/}, {\bf 146} (1992) pp. 454--481.

[22]
{\sc C.~Ehresmann}.
\newblock Sur la topologie de certains espaces {homog\`enes}.
\newblock {\em Ann. Math.\/}, {\bf 35} (1934) pp. 187--198.

[23]
{\sc S.~Fischer}.
\newblock {\em Signed poset homology and $q$-analog {M\"obius} functions\/}.
\newblock Ph.D. thesis, University of Michigan, Ann Arbor, MI (1993).

[24]
{\sc P.~C. Fishburn}.
\newblock {\em {Interval Graphs and Interval Orders}\/}.
\newblock Wiley, New York (1985).

[25]
{\sc M.~Geck and G.~Pfeiffer}.
\newblock {\em Characters of finite {C}oxeter groups and {I}wahori--{H}ecke
algebras\/}, vol.~21 of {\em London Mathematical Society Monographs. New
\newblock The Clarendon Press Oxford University Press, New York (2000).

[26]
{\sc I.~Gessel and G.~Viennot}.
\newblock Determinants and plane partitions (1989).
\newblock Preprint.

[27]
{\sc I.~P. Goulden and D.~M. Jackson}.
\newblock Immanants, {S}chur functions, and the {M}ac{M}ahon master theorem.
\newblock {\em Proc. Amer. Math. Soc.\/}, {\bf 115}, 3 (1992) pp. 605--612.

[28]
{\sc M.~Guay-Paquet}.
\newblock A second proof of the {Shareshian}--{Wachs} conjecture, by way of a
new {Hopf} algebra (2016).
\newblock Preprint {\tt math.CO/1601.05498} on {ArXiv}.

[29]
{\sc M.~Haiman}.
\newblock Hecke algebra characters and immanant conjectures.
\newblock {\em J. Amer. Math. Soc.\/}, {\bf 6}, 3 (1993) pp. 569--595.

[30]
{\sc M.~Harada, M.~Precup, and J.~Tymoczko}.
\newblock Toward permutation bases in the equivariant cohomology rings of
regular semisimple {Hessenberg} varieties.
\newblock {\em La Matematica\/}, {\bf 1} (2021) pp. 263--316.

[31]
{\sc F.~Harary}.
\newblock On the notion of balance of a signed graph.
\newblock {\em Michigan Math J.\/}, {\bf 2}, 2 (1953/1954) pp. 143--146.

[32]
{\sc P.~N. Hoefsmit}.
\newblock {\em Representations of Hecke algebras of finite groups with BN-pairs
of classical type\/}.
\newblock Ph.D. thesis, University of British Columbia, Vancouver, BC, Canada

[33]
{\sc J.~E. Humphreys}.
\newblock {\em Linear Algebraic Groups\/}, vol.~21 of {\em Graduate texts in
\newblock Springer-Verlag, New York (1975).

[34]
{\sc R.~Kaliszewski, J.~Lambright, and M.~Skandera}.
\newblock Bases of the quantum matrix bialgebra and induced sign characters of
the {H}ecke algebra.
\newblock {\em J. Algebraic Combin.\/}, {\bf 49}, 4 (2019) pp. 475--505.

[35]
{\sc S.~Karlin and G.~{McGregor}}.
\newblock Coincidence probabilities.
\newblock {\em Pacific J. Math.\/}, {\bf 9} (1959) pp. 1141--1164.

[36]
{\sc D.~Kazhdan and G.~Lusztig}.
\newblock {Representations of Coxeter groups and Hecke algebras}.
\newblock {\em Invent. Math.\/}, {\bf 53} (1979) pp. 165--184.

[37]
{\sc D.~Kazhdan and G.~Lusztig}.
\newblock {Schubert varieties and Poincar\'e duality}.
\newblock {\em Proc. Symp. Pure. Math., A.M.S.\/}, {\bf 36} (1980) pp.

[38]
{\sc M.~Konvalinka and M.~Skandera}.
\newblock Generating functions for {H}ecke algebra characters.
\newblock {\em Canad.\ J.\ Math.\/}, {\bf 63}, 2 (2011) pp. 413--435.

[39]
{\sc M.~Kuroda and S.~Tsujie}.
\newblock Chromatic signed-symmetric functions of signed graphs (2021).
\newblock Preprint {\tt math.CO/2101.03018} on {ArXiv}.

[40]
{\sc V.~Lakshmibai and B.~Sandhya}.
\newblock Criterion for smoothness of {Schubert} varieties in {$SL(n)/B$}.
\newblock {\em Proc. Indian Acad. Sci. (Math Sci.)\/}, {\bf 100}, 1 (1990) pp.

[41]
{\sc J.~Lambert}.
\newblock Theta-vexillary signed permutations.
\newblock {\em Electron. J. Combin.\/}, {\bf 25}, 4 (2018) pp. Paper No. 4.53,

[42]
{\sc A.~Lascoux and M.-P. Sch\"utzenberger}.
\newblock {Schubert Polynomials and the Littlewood-Richardson Rule}.
\newblock {\em Letters in Math. Physics\/}, {\bf 10} (1985) pp. 111--124.

[43]
{\sc B.~{Lindstr\"om}}.
\newblock On the vector representations of induced matroids.
\newblock {\em Bull. London Math. Soc.\/}, {\bf 5} (1973) pp. 85--90.

[44]
{\sc D.~E. Littlewood}.
\newblock {\em The {T}heory of {G}roup {C}haracters and {M}atrix
{R}epresentations of {G}roups\/}.
\newblock Oxford University Press, New York (1940).

[45]
{\sc I.~Macdonald}.
\newblock {\em {Symmetric Fuctions and Hall Polynomials}\/}.
\newblock Oxford University Press, Oxford (1979).

[46]
{\sc R.~Merris and W.~Watkins}.
\newblock Inequalities and identities for generalized matrix functions.
\newblock {\em Linear Algebra Appl.\/}, {\bf 64} (1985) pp. 223--242.

[47]
{\sc {OEIS Foundation Inc.}}
\newblock The {O}n-{L}ine {E}ncyclopedia of {I}nteger {S}equences.
\newblock Published electronically at {\tt http://oeis.org}.

[48]
{\sc M.~Precup and E.~Sommers}.
\newblock Perverse sheaves, nilpotent {Hessenberg} varieties, and the modular
\newblock {\em Pure Appl. Math. Q.\/} (To appear).

[49]
{\sc R.~A. Proctor}.
\newblock Classical {Bruhat} orders and lexicographic shellability.
\newblock {\em J. Algebra\/}, {\bf 77} (1982) pp. 104--126.

[50]
{\sc V.~Reiner}.
\newblock Signed posets.
\newblock {\em J.\ Combin.\ Theory Ser.\ A\/}, {\bf 62}, 2 (1993) pp. 324--360.

[51]
{\sc A.~Rok and A.~Szenes}.
\newblock {Eschers} and {Stanley's} chromatic $e$-positiviey conjecture (2023).
\newblock Preprint {\tt math.CO/2305.00963} on {ArXiv}.

[52]
{\sc J.~Shareshian} (2018).
\newblock Personal communication.

[53]
{\sc J.~Shareshian and M.~Wachs}.
\newblock Chromatic quasisymmetric functions and {Hessenberg} varieties.
\newblock In {\em Configuration Spaces\/} ({\sc A.~Bjorner, F.~Cohen, C.~D.
Concini, C.~Procesi, and M.~Salvetti}, eds.). Edizione Della Normale, Pisa
(2012), pp. 433--460.

[54]
{\sc J.~Shareshian and M.~L. Wachs}.
\newblock Chromatic quasisymmetric functions.
\newblock {\em Adv. Math.\/}, {\bf 295} (2016) pp. 497--551.

[55]
{\sc M.~Skandera}.
\newblock On the dual canonical and {Kazhdan}-{Lusztig} bases and 3412,
4231-avoiding permutations.
\newblock {\em J.\ Pure Appl.\ Algebra\/}, {\bf 212} (2008).

[56]
{\sc M.~Skandera}.
\newblock Characters and chromatic symmetric functions.
\newblock {\em Electron. J. Combin.\/}, {\bf 28}, 2 (2021).
\newblock Research Paper P2.19, 39 pages.

[57]
{\sc M.~Skandera}.
\newblock Generating functions for monomial characters of wreath products
$\mathbb{Z}/d\mathbb{Z} \wr \mathfrak{S}_n$.
\newblock {\em Enum. Combin. Appl.\/}, {\bf 1}, 2 (2021).
\newblock Paper S2R10, 10 pages.

[58]
{\sc R.~Stanley}.
\newblock {A symmetric function generalization of the chromatic polynomial of a
\newblock {\em Adv. Math.\/}, {\bf 111} (1995) pp. 166--194.

[59]
{\sc R.~Stanley}.
\newblock {\em {Enumerative Combinatorics}\/}, vol.~2.
\newblock Cambridge University Press, Cambridge (1999).

[60]
{\sc R.~Stanley}.
\newblock Positivity problems and conjectures.
\newblock In {\em Mathematics: Frontiers and Perspectives\/} ({\sc V.~Arnold,
M.~Atiyah, P.~Lax, and B.~Mazur}, eds.). American Mathematical Society,
Providence, RI (2000), pp. 295--319.

[61]
{\sc R.~Stanley and J.~R. Stembridge}.
\newblock {On immanants of Jacobi-Trudi matrices and permutations with
restricted positions}.
\newblock {\em J. Combin. Theory Ser. A\/}, {\bf 62} (1993) pp. 261--279.

[62]
{\sc J.~R. Stembridge}.
\newblock Ordinary representations of {$B_n$}.
\newblock Unpublished manuscript.

[63]
{\sc J.~R. Stembridge}.
\newblock Some conjectures for immanants.
\newblock {\em Canad.\ J.\ Math.\/}, {\bf 44}, 5 (1992) pp. 1079--1099.

[64]
{\sc H.~Tamvakis}.
\newblock Degeneracy locus formulas for amenable {Weyl} group elements (2019).
\newblock Preprint {\tt math.AG:1909.06398} on {ArXiv}.

[65]
{\sc W.~T. Trotter}.
\newblock {\em Combinatorics and Partially Ordered Sets: Dimension Theory\/}.
\newblock Johns Hopkins University Press, Baltimore (1992).

[66]
{\sc J.~S. Tymoczko}.
\newblock Permutation actions on equivariant cohomology of flag varieties.
\newblock In {\em Toric topology\/}, vol. 460 of {\em Contemp. Math.\/}. Amer.
Math. Soc., Providence, RI (2008), pp. 365--384.

[67]
{\sc J.~S. Tymoczko}.
\newblock Permutation representations on {S}chubert varieties.
\newblock {\em Amer. J. Math.\/}, {\bf 130}, 5 (2008) pp. 1171--1194.

[68]
{\sc T.~Zaslavsky}.
\newblock Signed graph coloring.
\newblock {\em Discrete Math.\/}, {\bf 39} (1982) pp. 215--228.

\end{thebibliography}

\end{document}

\section{Appendix?}

($\star$ Probably get rid of all of this?)

\begin{alg}\label{a:wtopq}
%applying the following single algorithm to
Given $w \in \bn$ \avoidingsignedp, do
%the following.
\begin{enumerate}
\item Let $p$ be the least positive letter
%greatest negative letter
%$[\ol n,\ol 1]$
in $w_1 \cdots w_n$
%    in $w_{\ol n} \cdots w_{\ol 1}$
%satisfying $w^{-1}_j < 0$
and define
$b_{\ol n} \cdots b_{\ol 1} b_1 \cdots b_n$ by
\begin{enumerate}
\item $b_i = w^{-1}_i$ for $i = \ol n, \dotsc, \ol p$,
\item $b_i = |i|$ for $i = \ol p +1 \dotsc, \ol 1$,
\item $b_i = w_i$ for $i = 1,\dotsc,n$.
\end{enumerate}
\item Define $m_{\ol n} \cdots m_{\ol 1} m_1 \cdots m_n$ by
$m_i = \max\{b_{\ol n}, \dotsc, b_i \}$.
\item For $i = 1,\dotsc, n-1$, define $i <_{Q(w)} m_i+1, \dotsc, n$.
%    $($where $\ol 1 + 1 \defeq 1)$. ($\star$ Is that necessary?)
\item For $i = 1,\dotsc, n$, if $w_i < 0$, circle element $|w_i|$.
%appears in the subword $w_{\ol n} \cdots w_{\ol 1}$,
%then place element $i$ in a box.
\end{enumerate}
\end{alg}
For example, consider the element
%$w = \ol521\ol3\ol443\ol1\ol25$
%  $w = \ol521\ol4\ol334\ol1\ol25$ in $\mfb5$.
$w = \ol31\ol22\ol13$ in $\mfb3$.
%Viewing $w$ as an element of $\mfs{[\ol5,5]}$ and
Using (\ref{eq:composition})
%to compute $F_w$, $P(w)$, $Q(w)$,
we construct
the fifth network
in (\ref{eq:bcdsn}),
the fifth unit interval order in (\ref{eq:bcuios}),
and the fifth type-$\msfBC$ unit interval order in ({\ref{eq:uioc}).
%  \begin{equation}
%    F_w = F'_{[2,4]} \bullet F'_{[1,3]} \bullet F'_{[\ol2,2]} =
%    \begin{tikzpicture}[scale=.5,baseline=-5]
%\node at (-.4,4.5) {$\scriptstyle{5}$};
%\node at (-.4,3.5) {$\scriptstyle{4}$};
%\node at (-.4,2.5) {$\scriptstyle{3}$};
%\node at (-.4,1.5) {$\scriptstyle{2}$};
%\node at (-.4,0.5) {$\scriptstyle{1}$};
%\node at (-.4,-0.5) {$\scriptstyle{\ol1}$};
%\node at (-.4,-1.5) {$\scriptstyle{\ol2}$};
%\node at (-.4,-2.5) {$\scriptstyle{\ol3}$};
%\node at (-.4,-3.5) {$\scriptstyle{\ol4}$};
%\node at (-.4,-4.5) {$\scriptstyle{\ol5}$};
%\node at (3.4,4.5) {$\scriptstyle{5}$};
%\node at (3.4,3.5) {$\scriptstyle{4}$};
%\node at (3.4,2.5) {$\scriptstyle{3}$};
%\node at (3.4,1.5) {$\scriptstyle{2}$};
%\node at (3.4,0.5) {$\scriptstyle{1}$};
%\node at (3.4,-0.5) {$\scriptstyle{\ol1}$};
%\node at (3.4,-1.5) {$\scriptstyle{\ol2}$};
%\node at (3.4,-2.5) {$\scriptstyle{\ol3}$};
%\node at (3.4,-3.5) {$\scriptstyle{\ol4}$};
%\node at (3.4,-4.5) {$\scriptstyle{\ol5}$};
%\draw[-] (0,4.5) -- (3,4.5);
%\draw[-] (0,3.5) -- (.5,2.5) -- (1, 3.5) -- (3,3.5);
%\draw[-] (0,2.5) -- (.5,2.5) -- (1.5, 1.5) -- (2,2.5) -- (3, 2.5);
%\draw[-] (0,1.5) -- (.5,2.5) -- (1.5,1.5) -- (2.5,0) -- (3,1.5);
%\draw[-] (0,0.5) -- (1,0.5) -- (1.5,1.5) -- (2.5,0) -- (3,0.5);
%\draw[-] (0,-0.5) -- (1,-0.5) -- (1.5,-1.5) -- (2.5,0) -- (3,-0.5);
%\draw[-] (0,-1.5) -- (.5,-2.5) -- (1.5,-1.5) -- (2.5,0) -- (3,-1.5);
%\draw[-] (0,-2.5) -- (.5,-2.5) -- (1.5,-1.5) -- (2,-2.5) -- (3,-2.5);
%\draw[-] (0,-3.5) -- (.5,-2.5) -- (1, -3.5) -- (3,-3.5);
%\draw[-] (0,-4.5) -- (3,-4.5);
%    P(w) =
%\draw[fill] (0,2) circle (1.2mm); \node at (.4,2) {$\scriptstyle{5}$};
%\draw[fill] (-1.5,1) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (-.5,1) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (.5,.25) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (1.5,.25) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (-1.5,-.25) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (-.5,-.25) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (.5,-1) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (1.5,-1) circle (1.2mm); %\node at (-.5,1) {$\scriptstyle{3}$};
%\draw[fill] (0,-2) circle (1.2mm); \node at (.4,-2) {$\scriptstyle{\ol5}$};
%\draw[-] (0,1) -- (0,0) -- (0,-1);
%    \qquad
%    Q(w) = .
%  \end{equation}
Using Algorithm~\ref{a:wtopq} on the other hand,
we see that $2$ is the least positive letter appearing in $w_1w_2w_3 = 2\ol13$
%the right half
%of $w$
and compute
\begin{equation*}
b_{\ol3} = w_{\ol3}^{-1} = \ol3, \quad
b_{\ol2} = w_{\ol2}^{-1} = \ol1.
\end{equation*}
Then we set $b_{\ol1} = |\ol1| = 1$,
%and $b_{\ol1} = |\ol1| = 1$.
we copy $b_1b_2 b_3 = w_1w_2 w_3$,
and we compute left-to-right records to obtain
%we have
\begin{equation}\label{eq:excompm}
%  \begin{gathered}
b_{\ol3} b_{\ol2} b_{\ol1} b_1 b_2 b_3 =
\ol3\ol112\ol13,\qquad
m_{\ol3} m_{\ol2} m_{\ol1} m_1 m_2 m_3 =
\ol3\ol11223.
%  \end{gathered}
\end{equation}
Increasing the letters $m_1m_2 = 22$ to $33$, we set
\begin{equation*}
1 <_Q 3,\quad 2 <_Q3.
\end{equation*}
Finally we circle $1$, since $\ol1$ appears in $w_1 w_2 w_3$,
and again arrive at the fifth type-$\msfBC$ unit interval order
in (\ref{eq:uioc}).

\begin{prop}\label{p:wtopqworks}
Given $w \in \bn$ \avoidingsignedp, the composition
(\ref{eq:composition}) agrees with
\end{prop}
\begin{proof}
$\star$ Do this.
Let $\pi = (\pi_{\ol n}, \dotsc, \pi_{\ol1}, \pi_1, \dotsc, \pi_n)$
be the unique path family of type $e$ covering $F_w$, and let $p$
be the greatest negative letter in $w_{\ol n} \cdots w_{\ol1}$.
We have ($\star$ By what previous result?)
that for $i \leq p$, path $\pi_i$ lies entirely below path $\pi_{i'}$
if and only if there is no $k \geq i'$ with $w_k < i$.
%  intersects path $\pi_j$
%  if $w_j \leq i$. Thus in the poset $P$,
%  $\pi_i$ can lie entirely below $\pi_j$ only if
%  $i$ is not less than
%incomparable to or greater than
%  $w^{-1}_k$ for all $k \leq i$.
In other words, if we define $b_j = w^{-1}_j$,
then $i$ is greater than or incomparable to
$\max \{ b_{\ol n}, \dotsc, b_{\ol i} \}$ in $P$; it is less than all other
We have ($\star$ by what previous result?)
that $\pi_{p+1},\dotsc, \pi_{\ol{p+1}}$ are pairwise intersecting.
Thus for $p+1 \leq i \leq \ol 1$, we have that $i$ is incomparable to
or greater than $\ol n, \dotsc, \ol{p+1}$ and less than $\ol{p+2},\dotsc,n$.

\end{proof}

The word $m_{\ol n} \dots m_{\ol1}m_1 \cdots m_n$ constructed by
Algorithm~\ref{a:wtopq} is closely related to the unique path family
$\pi = \{ \pi_{\ol n}, \dotsc, \pi_{\ol1}, \pi_1, \dotsc, \pi_n\}$
of type $e$ which covers $F_w$.
In particular, $m_i$ essentially counts paths $\pi_k$ which do not lie
entirely above
%touch or
%lie entirely below
$\pi_i$ in $P(\pi)$.
%essentially count
%Each number $m_i$ computed by the above algorithm essentially counts
%paths which touch or lie completely below
\begin{prop}\label{p:minterp}
Fix $w \in \bn$ \avoidingsignedp,
%let $F_w$ be the corresponding
%descending star network,
and let $P = P(w)$ be the corresponding
type-$\msfA$ unit interval order.
%, and , $P(w)$ as in
%  $\pi = \{ \pi_{\ol n}, \dotsc, \pi_{\ol1}, \pi_1, \dotsc, \pi_n\}$
%  be the unique path family of type $e$ covering $F_w$.
The word $m_{\ol n} \dots m_{\ol1}m_1 \cdots m_n$ constructed by the
Algorithm~\ref{a:wtopq} satisfies
$ |[\ol n, m_i]| = \# \{ k \,|\, k \not >_P i \}$.
%  \# \{ k \,|\, k \not \geq_P i \} =
%  \begin{cases} m_i + n + 1 &\text{if $i \leq \ol1$},\\
%    m_i + n &\text{if $i \geq 1$}.
%  \end{cases}
\end{prop}
\begin{proof}
$\star$ Do this.
\end{proof}

\begin{prop}\label{p:bchessword}
Fix $w \in \bn$ \avoidingsignedp.
The word $m_{\ol n} \dots m_{\ol1}m_1 \cdots m_n$ constructed by the above
algorithm satisfies
\begin{enumerate}
\item $m_i \geq i$ for $i \in [\ol n,n]$,
\item $m_{\ol n} \leq \cdots \leq m_{\ol1} \leq m_1 \leq \cdots \leq m_n$,
\item the integer partition
%    $(m_n+n,\dotsc,m_1+n,m_{\ol1}+n+1,\dotsc,m_{\ol n}+n+1)$
$(|[\ol n, m_n]|,\dotsc,|[\ol n, m_1]|,
|[\ol n, m_{\ol1}]|, \dotsc, |[\ol n, m_{\ol n}]|)$
is self-transpose.
\end{enumerate}
\end{prop}
\begin{proof} (1) Define $b_{\ol n} \cdots b_{\ol1} b_1 \cdots b_n$
and $j$ as in the algorithm.
%be the greatest negative letter appearing in
%  $w_{\ol n} \cdots w_{\ol1}$, and
Observe that the subset
%for $j \leq \ol1$, the set
\begin{equation*}
\{ b_{\ol n}, \dotsc, b_j \} = \{ w_{\ol n}^{-1}, \dotsc, w_j^{-1} \}
\end{equation*}
consists of $j - \ol n + 1$ distinct letters. Thus the
maximum of any $k$-element subset of these can be no smaller than
the $k$th smallest element of $[\ol n,n]$, and we have $m_i \geq i$ for
$i = \ol n, \dotsc, j$.  Then since $b_i \geq 1$ for $i = j+1,\dotsc,\ol1$,
we have $m_i \geq i$ for all negative indices.
%    Next, since $\{b_{j+1},\dotsc,b_{\ol1} \} = \{1, \dotsc, |j+1|\}$
%    we have that $m_1, \dotsc, m_{|j+1|}$ are all at least $|j+1|$.
Now consider the least index
%$k > j+1$
$k \geq 1$
for which we have $m_k = w_k < k$.
If $k = 1$ and $m_1 < 1$, then fewer than $n$ negative letters appear
in $w_{\ol n} \cdots w_{\ol 1}$, which implies that $b_{\ol 1} = 1$,
$m_{\ol 1} \geq 1$, and $m_1 \geq 1$, a contradictdion.
If $k > 1$, then
since $\{|w_1|, \cdots, |w_n|\} = [1,n]$,
at least one of the letters
$w_1 \cdots w_{k-1}$ must have absolute value $r \geq k$.
If that letter is positive, then we have
$m_k \geq r \geq k$, a contradiction.
If that letter is negative then the subword $r w_k$ of $w_1 \cdots w_n$
matches the signed pattern $\ol2 1$, another contradiction.

\noindent (2) The use of $\max$ in the formula for $m_i$ guarantees this.

\noindent (3) Represent the partition by the matrix
$A = (a_{i,j})_{i,j \in [\ol n,n]}$ having entries
\begin{equation}\label{eq:bcantiadj}
a_{i,j} = \begin{cases}
1 &\text{if $j \leq m_i$}\\
0 &\text{otherwise}.
\end{cases}
\end{equation}
Thus the collection of ones in $A$
forms the Young diagram of the desired partition,
and the number of ones in row $i$ equals $[\ol n, m_i]$,
Property (3) is equivalent to the claim that
this number
also equals the number of ones in column $\ol i$.

By Proposition~\ref{p:minterp},
%we have
%  \begin{equation*}
$[\ol n, m_i]$ equals $2n$ minus the number of elements in $P$ which are
greater than $i$.  By the definition of $A$, the number of ones in
column $j$ equals the number of elements $i$ with $a_{i,j} = 1$.
By Proposition~\ref{p:minterp},
this is $2n$ minus the number of elements less than $j$ in $P$.
Thus the number of ones in column $\ol i$ of $A$ equals
$2n$ minus the number of elements less than $\ol i$ in $P$.
By the self-duality of $P$, the claimed equality holds.
\end{proof}

%($\star$ Also, say something about how the formulation
%  of $m_{\ol n} \cdots m_n$ guarantees symmetry in the antiadjacency matrix of
%  the poset?

For example, consider again the element $\ol31\ol22\ol13$ of $\mfb 3$
and the word $m = \ol3\ol11223$ which clearly
%in (\ref{eq:excompm})
satisfies conditions (1) and (2) of the proposition.  To see that $m$ satisfies
condition (3), we
define the matrix $A$ as in (\ref{eq:bcantiadj}),
%compute the partition
%and Young diagram
the number of ones per row being
$(|[\ol3,\ol3]|, |[\ol3,\ol1]|, |[\ol3,1]|, |[\ol3,2]|, |[\ol3,2]|, |[\ol3,3]|)
%$(|[\ol3,3]|, |[\ol3,2]|, |[\ol3,2]|, |[\ol3,1]|, |[\ol3,\ol1]|, |[\ol3,\ol3]|)
%= (6,5,5,4,3,1)$:
= (1,3,4,5,5,6)$:
\begin{equation*}
A = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 1
\end{bmatrix}.
\end{equation*}
By inspection, $655431$ is self-conjugate.

%, \qquad
%and observe that its Young diagram is
%  in reverse
%and adding $333444$ we obtain the partition $655531$, whose Young diagram is
%  \tableau[scY]{\phn |
%  \phn, \phn, \phn |
%  \phn, \phn, \phn, \phn |
%  \phn, \phn, \phn, \phn, \phn |
%  \phn, \phn, \phn, \phn, \phn |
%  \phn, \phn, \phn, \phn, \phn, \phn |}\ .

\ssec{Old Hess}

By Theorem~\ref{t:catalan}, the number of

%for the regular semisimple type-$\msfBB$
%, or EC2, Problem 7.16(b), p. 452.?)
%There are more type-$\msfC$ Hessenberg words
%than there are elements
%Since both of these numbers grow faster than
%the number of elements $w_{\ol n} \cdots w_{\ol1} w_1 \cdots w_n$ of $\bn$

($\star$ Conclude that the type-$\msfBC$ graphs can't explain all
Hessenberg varieties.)
%$\ol2\ol1$, $\ol21$, $312$, $3\ol12$.
%$312$-avoiding permutations
%$w = w_1 \cdots w_n$ in $\sn$

However given such an element $w \in \bn$, we can assign a Hessenberg
word $m = m^{\msfC}(w)$ to it by the injective map
defined by steps (1) -- (2) of Algorithm~\ref{a:wtoq}.
%(1) -- (2).
%Proposition~\ref{p:algorithm} (1) -- (2).
($\star$ Need to explain why?)
($\star$ Or would it be better to be less specific about the relation
between $w$ and $m$?)

%These words
%Hessenberg words
%and corresponding bijectively to $312$-avoiding permutations
%$w_1 \cdots w_n \in \sn$
%$w \in \sn$
%by the map (\ref{eq:firsthess})
%  m_i = \max \{ w_1,\dotsc, w_i \}.
%(See (\ref{eq:firsthess}).)

%where $m$ is a type-$\msfBB$ or -$\sfC$ Hessenberg word,
%the generating
%function for these type-$\msfBC$ Frobenius characteristics,
%type-$\msfBC$ graph coloring, and/or

%($\star$ Pose questions.)

\begin{prob}
Find a family
%  \begin{equation*}
$\{ D_m \,|\, m \text{ a type-} \msfBB \text{ Hessenberg word } \}$
%\in \bn \ol2\ol1, \ol21, 312, 3\ol12 \}$
%\subseteq \hbnq
%  \end{equation*}
%      $\avoidsp $ \}$
of elements of $H^{\msfBC}_n(q)$
%  $\hbnq$
%(and a correspondence to a subset of the
%  \pavoiding elements of $\bn$?)
%with the property that
%  which satisfy
%  \begin{equation}\label{eq:BCtymXY}
%    \sum_{j=0} \ntksp \mathrm{ch}(\coh^{2j}(\chess(m(w))))q^j
%  =
such that the symmetric function
$\omega \YBCq(D_m)$ equals $\mathrm{Fr}^{\msfBB}(m;q)$.
%  \end{equation}
\end{prob}
\begin{prob}
Find a family
%  \begin{equation*}
$\{ D_w \,|\, w \in \bn \text{ \avoidssignedp} \}$
%  \text{ avoids the signed patterns }