%    \ol2\ol1, \ol21, 312, 3\ol12 \}$
%\subseteq \hbnq
%  \end{equation*}
%      $\avoidsp $ \}$
of elements of $H^{\msfBC}}_n(q)$
%(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
$\omega \YBCq(D_w)$ equals $\mathrm{Fr}^{\msfC}(m;q)$.
%  \end{equation}
\end{prob}
By Proposition~\ref{p:everysymmfnbcq},
such families must exist.
It would also be interesting to interpret
$\mathrm{Fr}^{\msfBB}(m;q)$ and $\mathrm{Fr}^{\sfC}(m;q)$
in terms of graph coloring.
\begin{prob}\label{p:chromB}
Define a
%n appropriate
$\mathcal G^{\msfBB} =
\{ G(m) \,|\, m \text{ a type-}\msfBB \text{ Hessenberg word } \}$
of graphs and a family of chromatic symmetric functions
$\{ X_{G,q}^{\msfBB} \,|\, G \in \mathcal G^{\msfBB} \}$
so that for each
type-$\msfBB$ Hessenberg word $m$,
%for each element $w \in \bn$ avoiding the signed patterns
%$\ol2\ol1$, $\ol21$, $312$, $3\ol12$,
%define a graph $G(w)$ and
%an appropriate type-$\msfBC$ chromatic symmetric function
%in $\mathbb Z[q] \otimes \Lambda_n(x,y)$
%and associate a graph $G = G(w)$ to each $\pavoiding$ ($\star$ ?)
%element $w \in \bn$
%so that
%which equals
%the left-hand side of (\ref{eq:BCtymXY})
%such that
the symmetric function
$X_{G(m),q}^{\msfBB}$ equals $\mathrm{Fr}^{\msfBB}(m;q)$.
%  (\ref{eq:gradedchBC}).
\end{prob}
\begin{prob}\label{p:chromC}
Define a
\begin{equation*}
\mathcal G^{\msfC} =
\{ G(w) \,|\, w \in \bn \text{ \avoidssignedp } \}
%\text{ avoids the signed patterns }
%\ol2\ol1, \ol21, 312, 3\ol12 \}
\end{equation*}
of graphs and a family of chromatic symmetric functions
$\{ X_{G,q}^{\msfC} \,|\, G \in \mathcal G^{\msfC} \}$
so that for each element $w \in \bn$
avoiding these signed patterns and type-$\msfC$ Hessenberg word
$m = m^{\msfC}(w)$,
%  type-$\mathsf B$ Hessenberg word $m$,
%for each element $w \in \bn$ avoiding the signed patterns
%$\ol2\ol1$, $\ol21$, $312$, $3\ol12$,
%define a graph $G(w)$ and
%an appropriate type-$\msfBC$ chromatic symmetric function
%in $\mathbb Z[q] \otimes \Lambda_n(x,y)$
%and associate a graph $G = G(w)$ to each $\pavoiding$ ($\star$ ?)
%element $w \in \bn$
%so that
%which equals
%the left-hand side of (\ref{eq:BCtymXY})
%such that
the symmetric function
$X_{G(w),q}^{\msfC}$ equals $\mathrm{Fr}^{\msfC}(m;q)$.
%  (\ref{eq:gradedchC}).
%  ($\star$ Mention a correspondence between $m$ and $w$.)
\end{prob}

($\star$ Mention what Abreu and Nigro said about the Lusztig variety here.
Is the Hessenberg variety a special case?)
%Mention that our definition could be the right one,
%or show that it is not.)

%($\star$ What type-$\mathsf B$ and type-$\mathsf C$ analogs of the following
%result can we conjecture/ask about?)
%  Fix $w$ avoiding the pattern $312$ with corresponding unit interval order $P$%,
%  and let $e$ be the number of edges in $\inc(P)$.
%  We have
%  \begin{equation*}
%    \sum_{j=0}^e \mathrm{ch}(H^{2j}(\mathcal H(w)))q^j = \omega X_{\inc(P),q}
%    = \omega Y_q(\wtc wq).
%  \end{equation*}
%($\star$ Ask for which element $g = g(w)$ the symmetric function
%$Y^{\mathsf B}_q(D)$ is equal to the sum of characteristics of cohomology
%pieces, as $\bn$-modules.)

\ssec{Path crossing lemma}

\begin{lem}\label{l:lowertouch}
Fix $w \in \bn$ \avoidingsignedp{} and let
$\pi = (\pi_{\ol n},\dotsc,\pi_{\ol 1},\pi_1,\dotsc,\pi_n)$
be the unique path family of type $e$ covering
the descending star network $F_w$. If indices $j \leq i \leq \ol 1$ satisfy
$w_i \leq j$, then paths $\pi_i$ and $\pi_j$ intersect.
\end{lem}

\section{Old BC unit interval orders}

%  In the zig-zag network $F_w$,

Moreovera union of a poset $P' = P'(w)$ on elements
$1,\dotsc, n$ and a poset $P'' = P''(w)$ on elements
$\{ \ol1,\dotsc, \ol n \}$, with an antiisomorphism of $P'$ and $P''$
given by $i \mapsto \ol i$.
The natural labeling of $P(w)$ and

in the literature.
The fact that $P(w)$ is a unit interval order implies more conditions
on positively and negatively labeled elements.
\begin{prop}\label{p:minantichain}
For $w \in \bn$ \avoidingp, define $P = P(w)$ and $P'(w)$ as above.
Let $1,\dotsc, k$ be complete set of minimal elements in $P'(w)$,
and let $i \geq 1$ be the minimum number such that $[\ol i, i]$
is not an antichain in $P$.
Then for $j = i, \dotsc, n$ we have
$\ol j <_P 1,\dotsc, n$ and $\ol n, \dotsc, \ol1 <_P j$.
\end{prop}
\begin{proof}
Reword the proposition and refer to earlier fact about unit interval orders?
Mention that this is Lemma 1.1 from Fischer thesis.
Remove or reinstate material from below?
Mention grounded paths and how they correspond to $[\ol i, i]$.
\end{proof}

%In particular, each poset $P(w)$, $w \in \bn$
%\item $i <_{P(w)} j$ implies $i < j$ (as integers),
%\item $i <_{P(w)} j$ if and only if $\ol j <_{P(w)} \ol i$,
%\item for all $i, j \in \{ k >0 \,|\, \ol k <_{P(w)} k \}$, we have $\ol i <_{P(w)} j$.
%Conditions (1), (2) are the defining conditions of type-$\msfC$ posets
%and condition (3) is the defining condition of (a relabeling of)
%signed posets in \cite[Lem.\,1.1]{SFischerThesis}.
%($\star$ Restate this.)

Let $F_w$
be the zig-zag network
corresponding to $w$ by the bijection following (\ref{eq:posetex}),
and let $\pi = (\pi_{\ol n}, \dotsc, \pi_{\ol1}, \pi_1, \dotsc, \pi_n)$
be the unique path family of type $e$ covering $F_w$.
%(Temporarily consider $w \in \snn$?)
%Each zig-zag network can be covered by a unique path family of type $e$.
%The paths in this family form a poset $P$ on $n$ elements
%when we
%When $\pi$ is the unique path family of type $e$ covering $F_w$,
%we define $P(w)$ to be the poset $Q(\pi)$.
%Each zig-zag network can be covered by a unique path family of type $e$.
%The paths in this family form a poset $P$ on $2n$ elements
%when we define $\pi_i < \pi_j$ by
%\item $i < j$ as integers,
%\item $\pi_i$ does not intersect $\pi_j$.
%Different descending star networks determine different posets,
%but two different zig-zag networks may determine the same poset.
By symmetry, $P(w)$
%each $2n$-element poset formed by a type-$\msfBC$
%descending star network
is a union of two $n$-element unit interval orders,
each dual
of the other.
%Let $R(w)$ and $R'(w)$ be the subposets $P(w)_{[n]}$ and $P(w)_{[\ol n,\ol1]}$,
Let $P' = P'(w)$ be the poset on elements $1, \dotsc, n$,
%$\pi_1, \dotsc, \pi_n$,
with $1, \dotsc, k$
minimal, and
let $P'' = P''(w)$ be the poset dual to $P'$
on elements $\ol n, \dotsc, \ol 1$,
%$\pi_{\ol n}, \dotsc, \pi_{\ol 1}$,
with $\ol k, \dotsc, \ol1$
%$\pi_{\ol k}, \dotsc, \pi_{\ol 1}$
All nonminimal elements of $P'$ are greater than all elements of $P''$,
and all nonmaximal elements of $P''$ are less than all elements of $P'$.
There are $k+1$ possibilities for comparabilities among the minimal-element
antichain in $P'$ and the maximal element antichain in $P''$.
In particular, for some $i$, $0 \leq i \leq k$, we have that
\begin{enumerate}
\item paths $\{\pi_{\ol i}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_i \}$
are grounded, by the definition following (\ref{eq:bczznotdsn}),
and therefore pairwise incomparable.
%to paths
%  each element in
%  $\{\pi_{\ol i}, \dotsc, \pi_{\ol 1} \}$
%  is incomparable to each element in
%  , with both sets empty if $i=0$,
\item nongrounded paths $\{ \pi_{i+1}, \dotsc, \pi_n \}$ are
greater than each element in $P''$, while
nongrounded paths $\{\pi_{\ol n}, \dotsc, \pi_{\ol{i+1}} \}$ are
%, also not grounded
less than each element in $P'$.
\end{enumerate}

Since $\ol3$ is the largest negative letter appearing in the left half
of $w$, we compute
\begin{equation*}
b_{\ol5} = w_{\ol5}^{-1} = \ol5, \quad
b_{\ol4} = w_{\ol4}^{-1} = \ol2, \quad
b_{\ol3} = w_{\ol3}^{-1} = \ol1.
\end{equation*}
Then $b_{\ol2} = |\ol2| = 2$ and $b_{\ol1} = |\ol1| = 1$.
Copying $b_1\cdots b_5 = w_1 \cdots w_5$ and computing left-to-right records
we have
\begin{equation*}
%  \begin{gathered}
b_{\ol5} \cdots b_{\ol1} b_1 \cdots b_5 =
\ol5\ol2\ol12134\ol1\ol25,\qquad
m_{\ol5} \cdots m_{\ol1} m_1 \cdots m_5 =
\ol5\ol2\ol12234445.
%  \end{gathered}
\end{equation*}
Increasing the letters $3444$ to $4555$, we set
\begin{equation*}
1 <_Q 4, 5,\quad 2 <_Q5,\quad 3 <_Q 5,\quad 4 <_Q 5.
\end{equation*}
Finally, we circle $1$ and $2$, since $\ol1$, $\ol2$ appear in $w_1 \cdots w_5$.

($\star$ Show poset.)

\begin{prop}\label{p:algorithm}
Fix $w \in \bn$ \avoidingsignedp.
%$\ol{21}$, $\ol21$, $312$, $3\ol12$.
One may construct the type-$\msfBC$ unit interval order $Q(w)$
as follows.
\end{prop}
%  We justify the validity of this algorithm in several steps.
\begin{proof}
%($\star$ Redo this completely.)
%To justify the second statement,
Clearly the word
$m_{\ol n} \cdots m_{\ol1} m_1 \cdots m_n$ is weakly increasing.
We claim that it also
%constructed in Step (2) of the algorithm
satisfies $m_i \geq i$ for all $i$.
%$i = \ol n, \dotsc, \ol 1, 1, \dotsc, n$.
To see this, observe that
$\{ b_{\ol n}, \dotsc, b_j \} = \{ w_{\ol n}^{-1}, \dotsc, w_j^{-1} \}$
consists of $j - \ol n + 1$ distinct numbers. 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 $b_i > 1$ 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 $|j+1|$.
($\star$ Now figure out why $m_i \geq i$ for $i = |j|,\dotsc,n$.)

Now define a poset $\widetilde P$ on $[\ol n,n]$ by
$i <_{\widetilde P} j$ if and only if $j > m_i$.
%    $2n \times 2n$
%    %$0$-$1$
%    matrix
%    $A = (a_{i,j})_{i,j \in [\ol n,n]}$
%    %in terms of $m_{\ol n} \cdots m_{\ol1} m_1 \cdots m_n$
%    by setting the entries in each row $i$ to be
%  \begin{equation*}
%    a_{i,\ol n} = \cdots = a_{i, m_i} = 1,
%    \qquad
%    a_{i, m_i+1} = \cdots = a_{i,m_n} = 0
%  \end{equation*}
%  and the remaining entries in row $i$ equal to $0$.
%  (where we interpret $a_{i,\ol1 + 1} \defeq a_{i,1}$).
%  in which row $i$ has ones in positions by
%Since $m_i \geq i$ for all $i$ and $m_{\ol n} \leq \cdots m_{\ol1} \leq m_1 \leq \cdots m_n$,
%  By the above, $A$ is the antiadjacency matrix (\ref{eq:antiadj}) ($\star$ Fix this.)
%\geq m_{i-1}$ This
%matrix in turn corresponds to
%  of a unit interval order
%  with elements labeled $\ol n, \dotsc, \ol 1, 1, \dotsc, n$.
We will show that $\widetilde P = P(w)$,
%this poset is the unit interval order $P(w)$:
the poset on
the unique path family
$\sigma = (\sigma_{\ol n}, \dotsc, \sigma_{\ol 1}, \sigma_1, \dotsc, \sigma_n)$
of type $e$ covering $F_w$.

%  First we claim that for $i \geq 1$ we have
%  $m_i = \max\{w_{\ol n}, \dotsc, w_i \}$.
%   For each positive index $i$, if $k = \max\{ w_1, \dotsc, w_i\}$,
%  then $i$, $k$ are incomparable in $P$ and $i <_P k+1$.
%  if $w_i > i$, then paths $\pi_i$ and
%  $\pi_{w_i}$ intersect.
%  type-$\msfBC$ unit
%  interval order $Q(w)$ constructed by the algorithm
%  first that since
%$w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n$ consists of distinct letters
%we have $m_i \geq i$ for all $i$.
%Now fix $i > 0$,
%let $\sigma$ be the unique path family of type $e$ covering $F_w$,
%and let
%Since there exists a path family
Let $\pi$
%= (\pi_{\ol n}, \dotsc, \pi_{\ol1}, \pi_1,\dotsc,\pi_n)$
be the unique path family of type $w$ covering $F_w$.
At each vertex where two paths of $\sigma$ intersect
(necessarily without crossing),
two corresponding paths of $\pi$ intersect and cross.

%The family $\pi$ is formed from $\sigma$ by
By the definition of $m_i$, we have some index $h \leq i$ satisfying
$w_h = m_i$.  (Is this true?)
Thus the path $\pi_h$ starts at source $h < i$
and terminates at sink $m_i \geq i$, crossing the path $\sigma_i$.
It follows that there exists a path in $F_w$ from source $i$ to sink
$m_i \geq i$, and by
%This path necessarily
\cite[Lem.\,3.5]{CHSSkanEKL}
that paths $\sigma_i$, $\sigma_{m_i}$ intersect
or are equal if $i = m_i$.
Thus in the poset $P(w)$, elements $i$ and $m_i$ are incomparable or equal.
%to $m_i$ in $P(w)$.
Now consider the path $\sigma_{m_i + 1}$ and assume that
it intersects path $\sigma_i$.  Then there is a path in $F_w$ from
source $i$ to sink $m_i+1$ and by
\cite[Obs.\,3.4]{CHSSkanEKL},
the intervals
and indices $i$, $m_i+1$ satisfy one of the conditions
\begin{enumerate}
\item one interval $[c_k,d_k]$ contains both $1$ and $m_i+1$,
%  $1$ and $m_i+1$ belong to a common interval
\item two intervals $[c_k,d_k] \prec [c_\ell,d_\ell]$ contain $1$ and $m_i+1$,
\end{enumerate}
In the first case,
%the interval can not satisfy $c_k > 1$, because
the intersection of $\sigma_i$, $\sigma_{m_i+1}$ in $F_{[c_k,d_k]}$ implies
that $\pi_i$, $\pi_{m_i+1}$ cross in $F_{[c_k,d_k]}$,
and that for some $h \leq i$ the path $\pi_h$ crosses both and terminates
at sink $d_k \geq m_i+1$.  This forces $w_h \geq m_i+1$, contradicting
that $w_h \leq m_i$.
%$\max\{w_{\ol n}, \dotsc, w_i \} = m_i$.
%Suppose therefore that the interval has the form $[\ol a, a]$.
In the second case, while there is a path in $F_w$ from source $i$ to sink
$m_i+1$, the paths $\sigma_i$, $\sigma_{m_i+1}$ do not cross.
Thus we have $i <_{P(w)} m_i+1$.
%in the poset $P(w)$, elements $i$ and $m_i$ are incomparable or equal.
Since no path $\sigma_{m_i+2},\dotsc,\sigma_n$ crosses $\sigma_{m_i+1}$
we also have $i <_{P(w)} m_i+2,\dotsc,n$.
\cite[Lem.\,3.5]{CHSSkanEKL} implies that
the condition $w_i = \ol a < 0$ for $i > 0$
(and necessarily $w_{\ol i} = a > 0$) is equivalent
to $\sigma_a$ intersecting $\sigma_{\ol a}$, and therefore to $a$ being
a circled element of $Q(w)$.
($\star$ Complete the second part of this proof
and try to shorten it and/or make it easier to understand.
Maybe state the cited Observation 3.4 and Lemma 3.5 in this paper?)
\end{proof}

\section{Graphical representation of products of KL basis elements in type BC}

A \msfBC-extension of Theorem~\ref{t:graphicalrepna}
requires a denominator more intricate than that appearing in

\begin{conj}\label{c:graphicalrepnbc}
Fix network $E = F'_{[c_1,d_1]} \circ \cdots \circ F'_{[c_t,d_t]}$,
and let $F$ be its condensed concatenation, as an element of
$\net A{[\ol n,n]}$. Show that for some
polynomial $p_E(q) \in \mathbb Z[q]$
%as in
%  (\ref{eq:pEq}).
%  For all $i,j$ with $[c_i,d_i] \precdot [c_j,d_j]$, let $m_{i,j}$
%  be the number of edges incident upon the interior vertices
%  of $F_{[c_i,d_i]}$ and $F_{[c_j,d_j]}$.
%    define the set
%        \begin{equation*}
%          \mathcal I \defeq
%          \{ [c_i,d_i] \,|\, c_i = \ol{d_i} \}
%          \cup
%          \{ [c_i,d_i] \,|\, c_i \geq 0 \}
%          \cup
%          \{ [\ol{c_i},\ol{d_i}] \,|\, c_i \geq 0 \},
%          \end{equation*}
%and define the relation $\precdot$ (\ref{eq:precdotdef}) on $\mathcal I$.
%For all $I,J \in \mathcal I$ with $I \precdot J$,
%let $m_{I,J}$
%        be the number of edges incident upon
%        the central vertices of the corresponding simple star networks,
%        %$F'_{[c_i,d_i]}$ and $F'_{[c_j,d_j]}$, and
%        and delete $m_{I,J} - 1$ of these to construct
%        $F = F'_{[c_1,d_1]} \bullet \cdots \bullet F'_{[c_t,d_t]} \in
%        \snet{BC}{[\ol n,n]}$.
%        by deleting $m_{i,j}-1$ of the above edges
the networks $E$ and $F$ graphically represent
%        Then $F$ graphically represents
%the element
\begin{equation}\label{eq:ratlcwbc}
\btc{s_{[c_1,d_1]}}q \cdots \btc{s_{[c_t,d_t]}}q, \qquad
%          \sum_{\pi \in \mathcal P(G)} q^{\dfct(\pi)} T_{\type(\pi)} =
\frac{\btc{s_{[c_1,d_1]}}q \cdots \btc{s_{[c_t,d_t]}}q}{p_E(q)}.
%               {\displaystyle{\prod_{I \prec \ntnsp \cdot J}}
%               {\displaystyle{\prod_{[c_i,d_i] \precdot [c_j,d_j]}} (m_{i,j})_q!}}
%                 (m_{I,J})_q!},
%                 \nTksp \nTksp \ntksp (m_{I,J})_q!},
%\in H_{[h,l]}^{\msfBC}(q).
\end{equation}
respectively, as elements of $\hbnq$.
\end{conj}

($\star$ Pose problem of stating a \msfBC-analog of Billey-Warrington result
for wiring diagrams.)

\section{Graphical representation of products of KL basis elements in type A}

Defining {\em $q$-analogs} of nonnegative integers and their factorials by
\begin{equation*}
n_q = \begin{cases}
1 + \cdots + q^{n-1} &\text{if $n>0$},\\
0 &\text{if $n=0$},
\end{cases} \qquad \qquad
n_q! = \begin{cases}
1_q \cdots n_q  &\text{if $n>0$},\\
1 &\text{if $n=0$},
\end{cases}
\end{equation*}
we have the following. (See also~\cite[Thm.\,4.3]{SkanNNDCB}.)
%      ($\star$  Mention factorization of KL basis elements too?)
%      \begin{lem}
%        Let star network $F$ graphically represent $g \in H_{[h,l]}(q)$.
%        Then $F \circ F_{[c,d]}$ graphically represents
%        $g \wtc{s_{[c,d]}}q$.
%      \end{lem}
%      \begin{proof}
%      \end{proof}
%      \begin{prop}
%        Let star networks $F$, $F'$ graphically represent
%        elements $g, g' \in H_{[h,l]}(q)$, and let
%        $F \bullet F'$ be obtained from $F \circ F'$ by the removal,
%        for $i=1,\dotsc,k$,
%        of $m_i$ edges between vertices $x_i \in F$, $y_i \in F'$.
%        Then
%        $F \circ F'$ graphically represents the product $gg'$,
%        and $F \bullet F'$ graphically represents the product
%        \begin{equation*}
%          \frac{gg'}{(m_1)_q! \cdots (m_k)_q!}.
%        \end{equation*}
%      \end{prop}
%      \begin{proof}
%        Just prove this from scratch.
%      \end{proof}
\begin{thm}\label{t:graphicalrepna}
Fix network $E = F_{[c_1,d_1]} \circ \cdots \circ F_{[c_t,d_t]}$.
%        and let $\precdot$ be the relation (\ref{eq:precdotdef}) defined on
%        \begin{equation*}
%          \mathcal I \defeq
%          \{ [c_i,d_i] \,|\, c_i = \ol{d_i} \}
%          \cup
%          \{ [c_i,d_i] \,|\, c_i \neq \ol{d_i} \}
%          \cup
%          \{ [\ol{c_i},\ol{d_i}] \,|\, c_i \neq \ol{d_i} \}.
%          \end{equation*}
%$I,J \in \mathcal I$ with $I \precdot J$, let $m_{I,J}$
For all $i,j$ with $[c_i,d_i] \precdot [c_j,d_j]$, let $m_{i,j}$
be the number of edges incident upon
the central vertices
%corresponding to these intervals, and
of $F_{[c_i,d_i]}$ and $F_{[c_j,d_j]}$, and
delete $m_{i,j} - 1$ of these to construct
$F = F_{[c_1,d_1]} \bullet \cdots \bullet F_{[c_t,d_t]} \in \snet A{[h,l]}$.
%        by deleting $m_{i,j}-1$ of the above edges
Then $E$ and $F$ graphically represent
%the elements
\begin{equation}\label{eq:ratlcw}
\wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_t,d_t]}}q, \qquad
%          \sum_{\pi \in \mathcal P(G)} q^{\dfct(\pi)} T_{\type(\pi)} =
\frac{\wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_t,d_t]}}q}
{\displaystyle{\prod_{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]}}
\nTksp \nTksp \ntksp (m_{i,j})_q!} \in H_{[h,l]}(q),
\end{equation}
respectively, as elements of $H_{[h,l]}(q)$.
\end{thm}
\begin{proof}
%        ($\star$ Probably remove this whole proof.)
%        The first statement was proved in
By \cite[Cor.\,5.3]{CSkanTNNChar}, $E$ graphically represents
the first element in (\ref{eq:ratlcw}).
%        \begin{equation}\label{eq:noratl}
%          \sum_\pi q^{\dfct(\pi)} T_{\type(\pi)} =
%          \wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_t,d_t]}}q,
%        \end{equation}
%        where the sum is over all path families $\pi$ which cover $E$.
Now choose indices $i < j$ with $[c_i,d_i] \precdot [c_j,d_j]$,
and fix a path family $\sigma = (\sigma_1,\dotsc,\sigma_n)$
covering $E$ such that on the $m \defeq m_{i,j}$ edges
the central vertices of $F_{[c_i,d_i]}$ and $F_{[c_j,d_j]}$,
we have paths $\sigma_{k_1}, \dotsc, \sigma_{k_m}$
with $k_1 < \cdots < k_m$ appearing in order from the
lowest edge to the highest edge.
%and $\sigma_{k_1}$ appearing below $\sigma_{k_2}, \dotsc,$
%appearing below $\sigma_{k_m}$.
Consider the set $S(\sigma)$ of path families covering $E$ which
agree with $\sigma$ on all but the $m$ edges above.
There are $m!$ such families, each corresponding to an element
%parametrized by a permutation
of $\mfs{\{k_1,\dotsc,k_m\}}$.
If $\rho \in S(\sigma)$ corresponds to $w \in \mfs{\{k_1,\dotsc,k_m\}}$
%with $\inv(w) = p$,
then we have $\dfct(\rho) = \dfct(\sigma) + \inv(w)$.
\begin{equation*}
\sum_{w \in \mfs{\ntnsp\{k_1,\dotsc,k_m\ntnsp\}}}
\nTksp \nTksp \ntksp q^{\inv(w)} = m_q!,
\end{equation*}
%it follows that
the removal from $E$ of $m-1$ of the edges above reduces
$\wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_t,d_t]}}q$
%$\sum_\pi q^{\dfct(\pi)}$
by a factor of $m_q!$.
\end{proof}
%      \begin{prop}
%        \end{prop}
%      ($\star$ Mention that this implies that for some $w \in \sn$,
%      we have a factorization of Kazhdan--Lusztig basis elements
%      of the form $r(q) \wtc{w^{(1)}}q \cdots \wtc{w^{(t)}}q$,
%      where $r(q)$ is a rational function of $q$.)
%Letting $\mathcal P(G)$ be the set of path families covering $G$,
%The connection between star networks and Kazhdan--Lusztig basis elements
%comes from

%Fix $v, w \in \mfs{[h,l]}$  with $w$ \avoidingp.
%  \begin{equation*}
%    \# \text{ path families of type $v$ covering } F_w =
%    \begin{cases}
%      1 &\text{if $v \leq w$},\\
%      0 &\text{otherwise}.
%    \end{cases}
%    \end{equation*}

%It is the unique Bruhat-maximal element of
%$\{ \type(\pi) \,|\, \pi \in \Pi(G) \}$.
%($\star$ Define path family?)
%($\star$ Mention that for all $i,j$ there is at most one path from source
%$i$ to sink $j$ in a zig-zag network?)

%Let $\pi = (\pi_h, \dotsc, \pi_l)$ be a path family
%covering a zig-zag network $F_w$,
%and define a poset $P = P(\pi)$ on $\{ \pi_h,\dotsc,\pi_l \}$ by
%declaring $\pi_i < \pi_j$ if $i < j$ and if the paths have no vertex in common.
%If $\sigma$ is the unique path family of type $e$ covering $F_w$,
%define $P(F_w)$ to be $P(\sigma)$.

%($\star$ Mention and cite relevant type-$\msfA$ results.)

%($\star$ Do I have to rethink the use of $i$, $j$, $k$, $m$ above?)

Certain star networks give us graphical representations of
Kazhdan--Lusztig basis elements.
%One significant

\section{Old Trace intro}

%Evaluating traces with cancellation-free formulas is not always easy.
Trace evaluation is not always easy, even in type $\msfA$.
(See, e.g., \cite[\S 1]{CHSSkanEKL}.)
But since traces are linear, one might hope to solve the
%In particular, we have the
following problem.

\bp\label{p:evaltrace}
Given a basis
%$\mathcal D =
$\{ D_w \,|\, w \in W \}$ of $H$,
and a basis
%$\Theta =
$\{ \theta_q^{(i)} \,|\, i = 1,\dotsc,p \}$
of $\trsp(H)$,
find combinatorial formulas for all evaluations
$\{ \theta_q^{(i)} (D_w) \,|\, i = 1,\dotsc,p, \ w \in W \}$.
%\theta_q^{(i)} \in \Theta, D_w \in \mathcal D\}$.
\ep
($\star$ Put this problem in the introduction?)
Type-$\msfA$ solutions were given in
[15], [34],
spanning sets consisting of products of Kazhdan--Lusztig basis elements
of the form $\smash{\wtc{s_{[i,j]}}q}$, and
the induced sign character basis of $\trsp(\hnq)$.
%and the
Partial type-$\msfA$ solutions were given in [14]
for the indepdendent set
$\{ \wtc wq \,|\, w \text{ \avoidsp} \}$
and various bases of $\trsp(\hnq)$.
Open special cases of Problem~\ref{p:evaltrace} include
%including the Shareshian--Wachs conjecture~\cite[Conj.\,4.9]{SWachsChromQ} that
%for all $w \in \sn$ \avoidingp, and
Haiman's conjecture~\cite[Conj.\,2.1]{HaimanHecke}
that $\phi_q^{\lambda}(\wtc wq) \in \mathbb N[q]$
%the same is true
for all $w \in \sn$.
This remains open even if we restrict $w \in \sn$ to \avoidp,
which by \cite[Thm.\,7.4]{CHSSkanEKL}
is equivalent to \cite[Conj.\,4.9]{SWachsChromQ}.
Also, the fact~\cite[Lem.\,1.1]{HaimanHecke}
that $\chi_q^\lambda(\wtc wq) \in \mathbb N[q]$ for all $w \in \sn$
has a combinatorial proof only in the case that $w$
\avoidsp~\cite[Thm.\,8.1]{CHSSkanEKL},~\cite[Thm.\,6.3]{SWachsChromQF}.
Thus it is interesting to study special cases of Problem~\ref{p:evaltrace}
even for subsets of the two bases.

\section{Old Intro}

In the interest of having a succinct introduction, we omit definitions of most
terms, saving them instead for the main body of the paper.

Let $\sn$ be the symmetric group,
let $\hnq$ be the Hecke algebra of type $\msfA$,
let $\trsp(\hnq)$ be the space of $\hnq$-traces,
for $w \in \sn$,
let $\wtc wq \defeq \smash{\qp{\ell(w)}2 C'_w(q)}$ be the
(modified, signless) Kazhdan--Lusztig basis element of $\hnq$ indexed by $w$.
%$w \in \sn$.
%Let $\trsp(\hnq)$ be the space of {\em $\hnq$-traces},
%the linear functions $\theta_q: \hnq \rightarrow \zqq$
%satisfing $\theta_q(hh') = \theta_q(h'h)$ for all $h, h' \in \hnq$.
%, let $\theta: \zsn \rightarrow \mathbb Z$
%denote the symmetric group trace obtained by specializing $\theta_q$ at
%$q = 1$.
%be a Hecke algebra trace which specializes at $q=1$ to the $\sn$ trace
%It is known that for
By \cite[Thm.\,4.3]{SkanNNDCB},
we have that for $w$ \avoidingp, there exists a planar network $F = F(w)$
which serves as a combinatorial interpretation for $\wtc wq$,
and which allows one to write a factorization of the form
\begin{equation*}
\wtc wq = \frac 1{f(q)} \wtc{w^{(1)}}q \cdots \wtc{w^{(r)}}q
\end{equation*}
with $f(q) \in \mathbb Z[q]$ and $(w^{(1)},\dotsc,w^{(r)})$ a sequence
of maximal elements of parabolic subgroups of $\sn$.
%Furthermore, we may associate to $w$
By \cite[Thm.\,7.4]{CHSSkanEKL},
there also exist
%a planar network $F = F(w)$,
a poset $P = P(w)$ and a graph $G = G(w)$
such that $F$, $P$, and $G$ aid in the combinatorial computation of
certain evaluations $\theta_q( \wtc wq )$ with
$\theta_q \in \trsp(\hnq)$.
%\hnq \rightarrow \zqq$
%an irreducible character, an induced trivial character,
%or an induced sign character,
%Examples of these
combinatorial computations involve
\begin{enumerate}
\item filling Young diagrams with paths in $F$,
\item filling Young diagrams with elements of $P$,
\item coloring vertices of $G$,
\item orienting edges of $G$,
\end{enumerate}
while obeying certain rules in each case.

\section{An observation about max,min letters in intervals appearing in a
zig-zag factorization}

When proving facts about zig-zag networks by induction, it is useful
to find a factor in (\ref{eq:bulletconcat}) whose removal creates
a smaller zig-zag network.  The following observation gives an easy way
to do so.
\begin{obs}\label{o:maxminint}
%It is easy to see that if
If $F \in \znet{A}{[h,l]}$
%is a zig-zag network of order $n$,
is formed from
%then at most one interval in
the concatenation (\ref{eq:bulletconcat}),
then the letters $h$ and $l$ each appear in
at most one of the intervals $[c_1,d_1],\dotsc,[c_t,d_t]$.
%in (\ref{eq:bulletconcat}).
%contains $l$, and at most one interval contains $h$.
An interval containing $h$ or $l$ must be
%and this interval must be
maximal or minimal (or both) in the partial order $\preceq$.
\end{obs}

\section{Old Hessenberg}

\ssec{Hessenberg varieties of type $\mathsf C$}

%  m_i = \max \{ w_1,\dotsc, w_i \}.
Write $\chess(m)$ for the regular semisimple type-$\msfC$ Hessenberg
variety associated to $m$.
It would be interesting to find a connection between
\begin{equation}\label{eq:gradedchC}
\sum_j \ntksp \mathrm{ch}(\coh^{2j}(\chess(m(w))))q^j,
%  = \omega Y_q(D_w).
\end{equation}
where $w \in \bn$ avoids the signed patterns
$\ol2\ol1$, $\ol21$, $312$, $3\ol12$,
%is a type-$\msfBB$ Hessenberg word,
%the generating
%function for these type-$\msfBC$ Frobenius characteristics,
%type-$\msfBC$ graph coloring, and/or
type-$\msfBC$ trace generating functions.

%($\star$ Pose questions.)

By Proposition~\ref{p:everysymmfnbcq},
such a family must exist.
It would also be interesting to interpret
(\ref{eq:gradedchC}) in terms of graph coloring.

%  Define an appropriate family of chromatic symmetric functions $X'_{G,q}$
%  so that
%  for each element $w \in \bn$ avoiding the signed patterns
%  $\ol2\ol1$, $\ol21$, $312$, $3\ol12$,
%  the symmetric function
%  $X'_{G(w),q}$ equals
%  (\ref{eq:gradedchC}).
%  \end{prob}

\begin{prob}
Find a family
%  \begin{equation*}
$\mathcal G^{\mathsf C}\{ D_w \,|\, w \in \bn  \}$
%\in \bn \ol2\ol1, \ol21, 312, 3\ol12 \}$
%\subseteq \hbnq
%  \end{equation*}
%      $\avoidsp $ \}$
of elements of $H^{\mathsf{BC}}_n(q)$
%(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
$\omega Y_q(D_m)$ equals
%  (\ref{eq:gradedchBC}).
%  \end{equation}
\end{prob}
By Proposition~\ref{p:everysymmfnbcq},
such a family must exist.
It would also be interesting to interpret
(\ref{eq:gradedchBC}) in terms of graph coloring.
\begin{prob}
Define a
%n appropriate
$\mathcal G^{\mathsf B} =
\{ G(m) \,|\, m \text{ a type-} \mathsf{B} \text{ Hessenberg word } \}$
of graphs and a family of chromatic symmetric functions
$\{ X'_{G,q} \,|\, G \in \mathcal G^{\mathsf B} \}$
so that for each
type-$\mathsf B$ Hessenberg word $m$,
%for each element $w \in \bn$ avoiding the signed patterns
%$\ol2\ol1$, $\ol21$, $312$, $3\ol12$,
%define a graph $G(w)$ and
%an appropriate type-$\msfBC$ chromatic symmetric function
%in $\mathbb Z[q] \otimes \Lambda_n(x,y)$
%and associate a graph $G = G(w)$ to each $\pavoiding$ ($\star$ ?)
%element $w \in \bn$
%so that
%which equals
%the left-hand side of (\ref{eq:BCtymXY})
%such that
the symmetric function
$X'_{G(m),q}$ equals
\end{prob}

Fix any diagonal matrix
$S \in \mathrm{SP}_{2n}(\mathbb C)$ having distinct nonzero eigenvalues,
the first $n$ of which have distinct absolute values,
and define the
{\em regular semisimple type-$\msfC$ Hessenberg variety associated to $m$}
%(or $P$, or $m_1 \cdots m_n$)
to be the set of cosets
%the subvariety
\begin{equation*}
\chess(m) = \chess(S,m) \defeq
\{ g\mathrm B \in \mathcal F^{\sfC}_n \,|\,
g \in \mathrm{SP}_{2n}(\mathbb C),\, g^{-1}Sg \in \HH^{\sfC}_m \}.
\end{equation*}

($\star$ Omit the following unless necessary:
and letting $\mathfrak b$
be its Lie algebra ($\star$ add precise description? Mention that these follow
Lakshmibai's conventions in Section 6 of her paper/book),
we have that conditions (1) -- (3) above are equivalent to the containments
\begin{equation*}
\mathfrak b \subseteq \HH^{\sfC}_m, \qquad [\mathfrak b, \HH^{\sfC}_m] \subseteq \HH^{\sfC}_m.
\end{equation*}

As in type $\msfA$,
the cohomology of $\chess(m)$ vanishes in odd degree,
\begin{equation*}
\coh^*(\mathcal \chess(m)) =
%  \bigoplus_{j=0}^{\ell(w_0w)}
\bigoplus_j
\coh^{2j}(\chess(m)),
\end{equation*}
%($\star$ compute correct maximum length and/or ask Juliana)
with each graded component having the structure of a $\bn$-module
via an action defined by Tymoczko~[66], [67].
($\star$ See Eric and Martha's paper)
($\star$ Was the action defined by
Balibanu and Crooks~\cite{BCPerverse}?)
Letting $\mathrm{ch}(\coh^{2j}(\chess(m)))$ be
the Frobenius characteristic of the character
of the module $\coh^{2j}(\chess(m))$, we may define
the generating function
\begin{equation}\label{eq:gradedchC}
\sum_j \ntksp \mathrm{ch}(\coh^{2j}(\chess(m(w))))q^j.
%  = \omega Y_q(D_w).
\end{equation}

It would be interesting to find a connection
between (\ref{eq:gradedchC})
%the generating
%function for these type-$\msfBC$ Frobenius characteristics,
%type-$\msfBC$ graph coloring, and/or
type-$\msfBC$ trace generating functions.

\section{signed pattern avoidance proofs}

($\star$ Decide what to do with this.)

By Theorem~\ref{t:dnetbcpavoid}, the special case of
Proposition~\ref{t:zigzagbc} in which
%$w \in \smash{\mfs{[h,l]}}$
$w$ avoids the signed patterns
$\ol2\ol1$, $\ol21$, $312$, $3\ol12$
yields a descending star network $F_w \in \dnet{BC}{[\ol n, n]}$.
%represents $\smash{\wtc wq}$.
This descending star network has a special relationship to the
other zig-zag networks in the theorem.
%~\cite[Thms.\,4.4, 5.6]{CHSSkanEKL}.
%In particular, we have the following
%we have the following.
\begin{proof}
(1) ($\star$ Wait.  Does this proof require posets, which will not be
introduced until the next section?  See Theorem~\cite[4.1]{CHSSkanEKL}.
Move this material to that section.)

(2) Fix $w \in \bn$ \avoidingp,
and let $F_w = F'_{[c_1,d_1]} \bullet \cdots \bullet F'_{[c_1,d_t]}$.
Let $u$ be the unique permutation in $\mfs t$ satisfying
$c_{u_1} > \cdots > c_{u_t}$.  Then the concatenation
$F'_{[c_{u_1},d_{u_1}]} \bullet \cdots \bullet F'_{[c_{u_t},d_{u_t}]}$ is a descending
star network $F_v$ for $v \in \bn$.  By Theorem~\ref{t:},
$v$ avoids the required signed patterns and by ? we have
that $Q(F_v) \cong Q(F_w)$.  By Theorem ?, $v$ is the only element
avoiding the required patterns in its equivalence class.

\end{proof}

\section{Maybe keep?}
%\section{Put this somewhere}

\ssec{$s$-length, $t$-length, and length}

($\star$ Is this necessary?)
We can decompose $\ell(w)$ as $\ell_s(w) + \ell_t(w)$, where
$\ell_s(w)$ and $\ell_t(w)$ are the numbers of generators of the forms
$s'_i$ and $t$ that appear in any reduced expression for $w$.
It is easy to read $\ell_t(w)$ from the one-line notation of $w$:
it is the number of negative letters in positions $1,\dotsc,n$,
equivalently, the number of negative letters
in the short one-line notation of $w$.
Note that $\ell_s(w)$ is not in general equal to $\ell(\phi(w))$.
For instance we have
\begin{equation*}
\ell_s(ts_1ts_1) = 2 \neq \ell(\phi(ts_1ts_1)) = 0.
\end{equation*}

%$\star$ Describe the two components of length: $\ell_s$ and $\ell_t$.)
\bp
($\star$ Is this necessary?)
Show that $\ell_s(w)$ is equal to the sum of the number of
{\em absolute value inversions},
(not necessarily consecutive subwords of the forms
$ba$, $\ol ba$, $b\ol a$, $\ol b \ol a$, with $|b| > |a|$)
and $2$ times the number of
{\em absolute value noninversions with negative second letter}
(not necessarily consecutive subwords of the forms
$a \ol b$, $\ol a \ol b$ with $|b| > |a|$).
\ep
\bp
($\star$ Is this important?)
How can we use the (short or long) one-line notation of $w$
to see if $s_i$ is a
left ascent ($s_iw > w$ in Bruhat order) or left descent ($s_iw < w$),
and a right ascent ($ws_i > w$) or right descent ($ws_i < s$)?
\ep

\section{Old $\tau$ path family}

\qquad
\begin{tikzpicture}[scale=.5,baseline=0]%s2s1s0
\node at (-2.1,3.5) {$\tau_4$};
\node at (-2.1,2.5) {$\tau_3$};
\node at (-2.1,1.5) {$\tau_2$};
\node at (-2.1,0.5) {$\tau_1$};
\node at (-2.1,-0.5) {$\tau_{\ol1}$};
\node at (-2.1,-1.5) {$\tau_{\ol2}$};
\node at (-2.1,-2.5) {$\tau_{\ol3}$};
\node at (-2.1,-3.5) {$\tau_{\ol4}$};
\draw[-, ultra thick] (-1.5,3.5) -- (0,2) -- (.5,2.5) -- (2.5,2.5);
\draw[-, ultra thick, dotted] (-1.5,2.5) -- (-.5,3.5) -- (2.5,3.5);
\draw[-, dashed]
(-1.5,1.5) -- (-.5,1.5) -- (0,2) -- (2,0) -- (2.5, .5);
\draw[-] (-1.5,0.5) -- (.5,0.5) -- (1.5,1.5) -- (2.5, 1.5);
\draw[-] (-1.5,-0.5) -- (.5,-0.5) -- (1.5, -1.5) -- (2.5,-1.5);
\draw[-, ultra thick, dashed]
(-1.5,-1.5) -- (-.5,-1.5) -- (0,-2) -- (2,0) -- (2.5,-.5);
\draw[-, ultra thick, dotted] (-1.5,-2.5) -- (-.5,-3.5) -- (2.5,-3.5);
\draw[-, ultra thick] (-1.5,-3.5) -- (0,-2) -- (.5,-2.5) -- (2.5,-2.5);
\end{tikzpicture}\,.

\section{Gavin's thesis}

This is true for two reasons. First, the equation
${\wtc{s_{[c,d]}}q}^2 = (d-c+1)_q!\wtc{s_{[c,d]}}q$ in $\hnq$ has analogs
\begin{equation*}
{\btc{s_{[c,d]}}q}^2 = \begin{cases}
(d-c+1)_q!\btc{s_{[c,d]}}q &\text{if $c > 0$},\\
(1+q)(1+q^2) \cdots (1+q^d) d_q! \btc{s_{[c,d]}}q  &\text{if $c = \ol d$}.
\end{cases}
\end{equation*}
in $\hbnq$.  For $k \geq 1$ let us define
\begin{equation*}
2_q^{\langle k \rangle} \defeq (1+q) (1+q^2) \cdots (1+q^k).
\end{equation*}
Second, a nontrivial type-$\msfBC$ simple star network (\ref{eq:csimplestars})
can have one or two internal vertices.
Thus for $[c_1,d_1] \cap [c_2,d_2] = k$,
the formation of
$F'_{[c_1,d_1]} \bullet F'_{[c_2,d_2]}$ from $F'_{[c_1,d_1]} \circ F'_{[c_2,d_2]}$
requires the removal of
one of the following sets of edges.
\begin{enumerate}
\item If $c_1 = \ol{d_1}$ and $c_2 = \ol{d_2}$, then
we remove $k-1$ edges between
the unique internal vertices of
$F'_{[c_1,d_1]}$ and $F'_{[c_2,d_2]}$.
\item If neither of the two intervals has the form $[\ol a, a]$, then
we remove $k-1$ edges between
the upper internal vertices of
$F'_{[c_1,d_1]}$ and $F'_{[c_2,d_2]}$, and
$k-1$ edges between
the lower internal vertices.
\item If exactly one of the two intervals has the form $[\ol a, a]$, then
we remove $k-1$ edges between
the unique internal vertex of $F'_{[\ol a,a]}$
and the upper internal vertex of the other simple star network,
and $k-1$ edges between
the unique internal vertex of $F'_{[\ol a,a]}$
and the lower internal vertex of the other simple star network.
\end{enumerate}
Given type-$\msfBC$
star network $E = F'_{[c_1,d_1]} \circ \cdots \circ F'_{[c_t,d_t]}$,
and partial order $\preceq$ on the $t$ intervals,
we define ($\star$ something like) the polynomial
\begin{equation}\label{eq:pEq}
p_E(q) \defeq
\nTksp \nTksp \prodsb{[c_i,d_i] {\prec \ntnsp \cdot} [c_j,d_j]\\ c_i = \ol d_i,\ c_j = \ol d_j} \nTksp \nTksp
2_q^{\langle \min\{d_i,d_j\}\rangle} (\min\{d_i,d_j\})_q!
\ \cdot \nTksp \nTksp \prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i \geq 1 \text{ or } c_j \geq 1} \nTksp \nTksp
(|[c_i,d_i] \cap [c_j,d_j]| - 1)_q!.
\end{equation}
%($\star$ Why is $\precdot$ not working?)

%($\star$ Rewrite this so that it might be correct, keeping in mind that
%  \btc{s'_{[\ol3,3]}}q \btc{s'_{[\ol3,3]}}q
%  =
%%  (1+q)(1+q^2)(1+q^3)3_q! \btc{s'_{[\ol3,3]}}q
%  = 2_q 2_{q^2} 2_{q^3} 3_q! \btc{s'_{[\ol3,3]}}q,
%i.e., that repeated edges between $F_{[\ol k, k]}$ and $F_{[\ol m,m]}$
%mean something different than between $F_{[\ol k, k]}$ and $F_{[a,b]}$
%or $F_{[a,b]}$ and $F_{[c,d]}$, assuming $a,c>0$.)
%($\star$ Define $2_q^{\langle k \rangle} = (1+q) (1+q^2) \cdots (1+q^k)$ ?)
%($\star$ Have Gavin define what it means for a type-$\msfBC$ zig-zag
%network in $\znet{BC}{[\ol n,n]}$
%to {\em graphically represent} an element of $\hbnq$. Include
%the correct generalization of type-$\msfA$ defects.)
%($\star$ Have Gavin state the following correctly and prove.)

($\star$ Fix the above polynomial: what if we have
$F'_{[\ol 8, 8]} \circ F'_{[2,3]} \circ F'_{[6,9]} \circ F'_{[\ol 6, 6]}$
and we delete edges to form
$F'_{[\ol 8, 8]} \bullet F'_{[2,3]} \bullet F'_{[6,9]} \bullet F'_{[\ol 6, 6]}$?
Then the six paths shared by the first and last simple network are at ``levels'' $1$, $4$, $5$.  Does the deletion count for $(1+q)(1+q^4)(1+q^5)$
or $2_q^{\langle 3\rangle}$?)

Now we conjecture the following $\msfBC$-extension of
%($\star$ Or should this be left as an open problem?)

\section{integer partitions}

Conjugacy classes of $\sn$ correspond to {\em integer partitions of $n$},
%to be a
%The number of conjugacy classes of $\sn$ equals
%the number of {\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 $n$.
Sometimes we use the notation $k^{a_k}$ to denote a
sequence of $a_k$ copies of the letter $k$.
Given $\lambda \vdash n$, we define the {\em transpose}
partition $\lambda^\tr = (\lambda_1^\tr, \dotsc,\lambda_{\lambda_1}^{\tr})$ by
$\lambda_i^\tr = \# \{ j \,|\, \lambda_j \geq i \}$. Thus $n^\tr = 1^n$.
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 the set of permutations
%by mapping each per
%Each conjugacy class of $\sn$ consists of permutations
whose {\em cycle type}, the weakly decreasing sequence of its
cycle cardinalities, is equal to $\lambda$.
Let $\ctype(w)$ denote the cycle type of $w \in \sn$.

%($\star$ Do we need/want the scalars of $\trsp(\hnq)$ to be $\zqq$
%or $\mathbb Z$?)
%The $\zqq$-module $\trsp(\hnq)$ also has rank equal to the number
%of integer partitions of $n$, and consists of functions
%$\theta_q: T_w \mapsto f(q)$
%which specialize at $\qp12 = 1$
%to elements $\theta = \theta_1$
%: w \mapsto f(1) \in \mathbb Z$
%of $\trsp(\sn)$ satisfying $\theta(w) = f(1)$.

%\ssec{Long and short one-line notations in $\bn$}
%Thus we define the one-line notation of $w \in \bn$ to be
%letting $s_i$, $t$ act on the
%  \begin{gathered}
%    s_i(\ol n \cdots \ol 1 1 \cdots n) =
%    \ol n \cdots \ol{(i+2)} \ol i \ol {(i+1)} \ol{i-1} \cdots \ol 1
%    1 \cdots (i-1) (i+1) i (i+2) \cdots n,\\
%    t(\ol n \cdots \ol 1 1 \cdots n) =
%    \ol n \cdots \ol 2 1 \ol 1 2 \cdots n,
%  \end{gathered}
%and by defining the one-line notation of $s_{i_1} \cdots s_{i_r}$ to be
%  s'_{i_1}(s'_{i_2}( \cdots (s'_{i_r}( \ol n \cdots \ol 1 1 \cdots n)) \cdots )),
%and w

%We can similarly replace the long functional one-line notation
%$w(\ol n) \cdots w(\ol 1) w(1) \cdots w(n)$ with short functional
%one-line notation $w(1) \cdots w(n)$.

Conjugacy classes of $\bn$ correspond to {\em bipartitions of $n$},
%to be a
%The number of conjugacy classes of $\sn$ equals
%the number of {\em integer partitions} of $n$,
pairs $(\lambda,\mu)$ of partitions satisfying $|\lambda| + |\mu| = n$.
We let the expression
$(\lambda,\mu) \vdash n$ denote that $(\lambda,\mu)$ is
a bipartition of $n$.
To explicitly describe the conjugacy classes of $\bn$,
we define the homomorphism
\begin{equation}\label{eq:varphi}
\begin{aligned}
\varphi: \bn &\rightarrow \sn \\
s'_i &\mapsto s_i, \quad i=1,\dotsc,n-1,\\
t &\mapsto e,
\end{aligned}
%  \begin{gathered}
%    \varphi: \bn \rightarrow \sn \\
%    s'_i \mapsto s_i, \qquad
%    t \mapsto e,
%  \end{gathered}
\end{equation}
%denote the homomorphism mapping $s'_i \mapsto s_i$, $t_i \mapsto e$.
%Equivalently, $\varphi$
which maps each letter in the short one-line notation of
$v \in \bn$ to its absolute value.
For each element $v \in \bn$
%$(v, \varphi(v)) \in \bn \times \sn$
and each cycle $C = (a_1, \dotsc, a_k)$
%(of positive letters)
of $\varphi(v) \in \sn$,
define the {\em signed cycle} $\hat C = (\hat a_1, \dotsc, \hat a_k = \hat a_0)$
of $v$ by
\begin{equation*}
\hat a_i = \begin{cases}
a_i &\text{if $v(a_{i-1}) = a_i$},\\
\ol a_i &\text{if $v(a_{i-1}) = \ol{a_i}$}.
\end{cases}
\end{equation*}

($\star$ Is this equivalent to saying that
$\hat a_i = a_i$ if $a_i$ appears in $v_1 \cdots v_n$,
and $\hat a_i = \ol{a_i}$ if $\ol{a_i}$ appears in $v_1 \cdots v_n$?)

%where $a_0 \defeq a_k$.
Call $\hat C$ {\em positive} if it has an even number of negative letters,
and {\em negative} otherwise.
The conjugacy class of $\bn$ corresponding to $(\lambda,\mu) \vdash n$
is precisely the set of elements
%by mapping each per
%Each conjugacy class of $\sn$ consists of permutations
whose {\em signed cycle type}, the bipartition of
%pair of weakly decreasing sequences of its
positive cycle cardinalities and negative cycle cardinalities,
is equal to $(\lambda,\mu)$.
We write $\sct(w) = (\lambda,\mu)$.

%The conjugacy class of $\bn$ corresponding to $(\lambda,\mu) \vdash n$
%is the set of elements
%%by mapping each per
%%Each conjugacy class of $\sn$ consists of permutations
%whose {\em signed cycle type} equals $(\lambda,\mu)$,
%where , the weakly decreasing sequence of its
%cycle cardinalities, is equal to $\lambda$.

\ssec{Transpositions}

In both $\snn$ and $\bn$, we define a {\em reflection} to be
any element which is conjugate to a generator.
Every reflection in $\snn$ has the form $(i,j)$, i.e., its action
on $w_{\ol n} \cdots w_n$ swaps the letters in positions $i$ and $j$.
($\star$ Maybe mention?  Note that while reflections in $\bn$
necessarily belong to $\snn$,
they are not in general reflections in $\snn$.)

\begin{lem}\label{l:transpolengths}
For all $i,j \in [\ol n, n]$,
the transposition $(i,j)$ has odd length.
%If $j \neq \ol i$, then
In particular, assuming $i < j$, we have
\begin{equation*}
( \ell_s(i,j), \ell_t(i,j) ) = \begin{cases}
(  2(j-i) - 1, 0 ) &\text{if $0 < i < j$},\\
( 2j + 2i - 3, 2) &\text{if $i < 0 < j$},\\
( 2i-2, 1) &\text{if $j = \ol i$}.
\end{cases}
\end{equation*}
\end{lem}
\begin{proof}
We have
\begin{equation*}
(i,j) = \begin{cases}
s'_{j-1} \cdots s'_{i+1}s'_is'_{i+1} \cdots s'_{j-1}
&\text{if $0 < i < j$},\\
%      we have
%  \begin{equation*}
%    (i,j) =
s'_{j-1} \cdots s'_1 t s'_1 s'_2 \cdots s'_{\ol i}  \cdots s'_2 s'_1 t s'_1 \cdots s'_{j-1}
&\text{if $i < 0 < j$},\\
s'_{i-1} \cdots s'_1 t s'_1 \cdots s'_{i-1}
&\text{ if $j = \ol i$}.
\end{cases}
\end{equation*}
\end{proof}

\ssec{The $\varphi$ homomorphism}

($\star$ Necessary?)
Some aspects of $\bn$ can be understood via its relation to $\sn$.
Define a homomorphism
\begin{equation*}
\begin{aligned}
\varphi: \bn &\rightarrow \sn\\
t &\mapsto 1,\\
s_i' &\mapsto s_i \, \text{ for $i \geq 1$}.
\end{aligned}
\end{equation*}

($\star$ Is this necessary?)
Applying the homomorphism $\varphi$ to long and short one-line notation,
we have
\begin{equation*}
\varphi(w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n) = |w_1| \cdots |w_n|,
\quad \varphi(w_1 \cdots w_n) = |w_1| \cdots |w_n|,
%\varphi(w(\ol n) \cdots w(\ol 1) w(1) \cdots w(n)) = |w(1)| \cdots |w(n)|,
%\quad \varphi(w(1) \cdots w(n)) = |w(1)| \cdots |w(n)|?
\end{equation*}

\ssec{Maybe not useful?}

\begin{prop}
If $\hat G$ is a type-$\msfBC$ descending star network and $w = w(\hat G)$,
then the type-$B$ and type-$C$ Schubert varieties
indexed by $w$ are both smooth.
\end{prop}
\begin{proof}
(Idea) The avoidance of the signed patterns
$\ol2\ol1$, $\ol21$, $312$, $3\ol12$
implies avoidance of \theps and therefore
guarantees rational (?)
smoothness of corresponding Schubert varieties in types $B$ and $C$.
\end{proof}

\ssec{Path matrices of star networks of type $BC$}

To each type-$\msfA$ zig-zag network
$\hat G$, we may associate a matrix
$A = A(\hat G) = (a_{i,j})_{i,j \in [\ol n, n]}$ by
\begin{equation*}
a_{i,j} = \begin{cases} 1 &\text{if there is a path in $\hat G$
from source $i$ to sink $j$},\\
0 &\text{otherwise.}
\end{cases}
\end{equation*}

Given a star network $G$ of type $BC$, define the $2n \times 2n$ path matrix
$A = A(D) = (a_{i,j})$ by
%(a_{i,j})_{i,j \in [\ol n,1] \cup [1,n]}$ by
\begin{equation*}
a_{i,j} = \# \text{ paths from source $i$ to sink $j$}.
\end{equation*}
For example, the three type-$\msfBC$ star networks
\begin{equation}\label{eq:3nets}
\begin{gathered}
\begin{tikzpicture}[scale=.5,baseline=0]
\node at (0,3.5) {$(4)$};            %\node at (1,1.5) {$2$};
\node at (0,2.5) {$(3)$};            %\node at (1,5.5) {$n$};
\node at (0,1.5) {$(2)$};          %\node at (1.75,4.5) {$n-1$};
\node at (0,0.5) {$(1)$};            %\node at (1,1.5) {$2$};
\node at (0,-0.5) {$(\ol 1)$};            %\node at (1,0.5) {$1$};
\node at (0,-1.5) {$(\ol 2)$};       %\node at (1,-0.5) {$\ol 1$};
\node at (0,-2.5) {$(\ol 3)$};       %\node at (1,-1.5) {$\ol 2$};
\node at (0,-3.5) {$(\ol 4)$};            %\node at (1,0.5) {$1$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (-1.5,3.5) -- (-.5,3.5);
\draw[-] (-1.5,2.5) -- (-.5,0.5);
\draw[-] (-1.5,1.5) -- (-.5,1.5);
\draw[-] (-1.5,0.5) -- (-.5,2.5);
\draw[-] (-.5,3.5) -- (.5,2.5);
\draw[-] (-.5,2.5) -- (.5,3.5);
\draw[-] (-.5,1.5) -- (.5,1.5);
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (.5,3.5) -- (1.5,3.5);
\draw[-] (.5,2.5) -- (1.5,1.5);
\draw[-] (.5,1.5) -- (1.5,2.5);
\draw[-] (.5,0.5) -- (1.5,0.5);
\draw[-] (-1.5,-3.5) -- (-.5,-3.5);
\draw[-] (-1.5,-2.5) -- (-.5,-0.5);
\draw[-] (-1.5,-1.5) -- (-.5,-1.5);
\draw[-] (-1.5,-0.5) -- (-.5,-2.5);
\draw[-] (-.5,-3.5) -- (.5,-2.5);
\draw[-] (-.5,-2.5) -- (.5,-3.5);
\draw[-] (-.5,-1.5) -- (.5,-1.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\draw[-] (.5,-3.5) -- (1.5,-3.5);
\draw[-] (.5,-2.5) -- (1.5,-1.5);
\draw[-] (.5,-1.5) -- (1.5,-2.5);
\draw[-] (.5,-0.5) -- (1.5,-0.5);
\end{tikzpicture}\\
G'_{\smash{[1,3]}} \bullet G'_{\smash{[3,4]}} \bullet G'_{\smash{[2,3]}}
\end{gathered}
\ ,
\qquad \qquad
\begin{gathered}
\begin{tikzpicture}[scale=.5,baseline=0]
\node at (0,3.5) {$(4)$};            %\node at (1,1.5) {$2$};
\node at (0,2.5) {$(3)$};            %\node at (1,5.5) {$n$};
\node at (0,1.5) {$(2)$};          %\node at (1.75,4.5) {$n-1$};
\node at (0,0.5) {$(1)$};            %\node at (1,1.5) {$2$};
\node at (0,-0.5) {$(\ol 1)$};            %\node at (1,0.5) {$1$};
\node at (0,-1.5) {$(\ol 2)$};       %\node at (1,-0.5) {$\ol 1$};
\node at (0,-2.5) {$(\ol 3)$};       %\node at (1,-1.5) {$\ol 2$};
\node at (0,-3.5) {$(\ol 4)$};            %\node at (1,0.5) {$1$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (-1.5,3.5) -- (-.5,3.5);
\draw[-] (-1.5,2.5) -- (-.5,2.5);
\draw[-] (-1.5,1.5) -- (-.5,-1.5);
\draw[-] (-1.5,0.5) -- (-.5,-0.5);
\draw[-] (-.5,3.5) -- (.5,2.5);
\draw[-] (-.5,2.5) -- (.5,3.5);
\draw[-] (-.5,1.5) -- (.5,1.5);
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (.5,3.5) -- (1.5,3.5);
\draw[-] (.5,2.5) -- (1.5,1.5);
\draw[-] (.5,1.5) -- (1.5,2.5);
\draw[-] (.5,0.5) -- (1.5,0.5);
\draw[-] (-1.5,-3.5) -- (-.5,-3.5);
\draw[-] (-1.5,-2.5) -- (-.5,-2.5);
\draw[-] (-1.5,-1.5) -- (-.5,1.5);
\draw[-] (-1.5,-0.5) -- (-.5,0.5);
\draw[-] (-.5,-3.5) -- (.5,-2.5);
\draw[-] (-.5,-2.5) -- (.5,-3.5);
\draw[-] (-.5,-1.5) -- (.5,-1.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\draw[-] (.5,-3.5) -- (1.5,-3.5);
\draw[-] (.5,-2.5) -- (1.5,-1.5);
\draw[-] (.5,-1.5) -- (1.5,-2.5);
\draw[-] (.5,-0.5) -- (1.5,-0.5);
\end{tikzpicture}\\
G'_{\smash{[\ol2,2]}} \bullet G'_{\smash{[3,4]}} \bullet G'_{\smash{[2,3]}}
\end{gathered}
\ ,
\qquad \qquad
\begin{gathered}
\begin{tikzpicture}[scale=.5,baseline=0]
\node at (0,3.5) {$(4)$};            %\node at (1,1.5) {$2$};
\node at (0,2.5) {$(3)$};            %\node at (1,5.5) {$n$};
\node at (0,1.5) {$(2)$};          %\node at (1.75,4.5) {$n-1$};
\node at (0,0.5) {$(1)$};            %\node at (1,1.5) {$2$};
\node at (0,-0.5) {$(\ol 1)$};            %\node at (1,0.5) {$1$};
\node at (0,-1.5) {$(\ol 2)$};       %\node at (1,-0.5) {$\ol 1$};
\node at (0,-2.5) {$(\ol 3)$};       %\node at (1,-1.5) {$\ol 2$};
\node at (0,-3.5) {$(\ol 4)$};            %\node at (1,0.5) {$1$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (-1.5,3.5) -- (-.5,3.5);
\draw[-] (-1.5,2.5) -- (-.5,2.5);
\draw[-] (-1.5,1.5) -- (-.5,1.5);
\draw[-] (-1.5,0.5) -- (-.5,-0.5);
\draw[-] (-.5,3.5) -- (.5,2.5);
\draw[-] (-.5,2.5) -- (.5,3.5);
\draw[-] (-.5,1.5) -- (.5,1.5);
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (.5,3.5) -- (1.5,3.5);
\draw[-] (.5,2.5) -- (1.5,1.5);
\draw[-] (.5,1.5) -- (1.5,2.5);
\draw[-] (.5,0.5) -- (1.5,0.5);
\draw[-] (-1.5,-3.5) -- (-.5,-3.5);
\draw[-] (-1.5,-2.5) -- (-.5,-2.5);
\draw[-] (-1.5,-1.5) -- (-.5,-1.5);
\draw[-] (-1.5,-0.5) -- (-.5,0.5);
\draw[-] (-.5,-3.5) -- (.5,-2.5);
\draw[-] (-.5,-2.5) -- (.5,-3.5);
\draw[-] (-.5,-1.5) -- (.5,-1.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\draw[-] (.5,-3.5) -- (1.5,-3.5);
\draw[-] (.5,-2.5) -- (1.5,-1.5);
\draw[-] (.5,-1.5) -- (1.5,-2.5);
\draw[-] (.5,-0.5) -- (1.5,-0.5);
\end{tikzpicture}\\
G'_{\smash{[\ol1,1]}} \bullet G'_{\smash{[3,4]}} \bullet G'_{\smash{[2,3]}}
\end{gathered}
\quad
\end{equation}
have path matrices
\begin{equation*}%\label{eq:3matrices}
\qquad \begin{bmatrix}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\
1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\
1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{bmatrix},
\quad
\begin{bmatrix}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{bmatrix},
\quad
\begin{bmatrix}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{bmatrix},
\end{equation*}

\section{$A^+$, $A^-$ examples}

For each planar network $G$ in $\net{A}{[h,l]}$,
define its {\em path matrix} $A = A(D) = (a_{i,j})_{i,j \in [h,l]}$ by
\begin{equation*}
a_{i,j} = \# \text{paths from source $i$ to sink $j$ in $G$}.
\end{equation*}
For example, the star network
$G_{[\ol 2,1]} \circ G_{[\ol 1,2]} \circ G_{[\ol 2,1]} \in \net{A}{[\ol2,2]}$
in (\ref{eq:allbutone})
has path matrix $A$ and related matrices $A^+$, $A^-$ (defined by (\ref{eq:pm}))
given by
%$G_{23\ol1}$ in (\ref{eq:dsn1})
%corresponding to $23\ol1 \in \mfb{3}$
%has path matrix
\begin{equation*}
A = \begin{bmatrix}
5 & 5 & 5 & 2 \\
5 & 5 & 5 & 2 \\
5 & 5 & 5 & 2 \\
2 & 2 & 2 & 1
\end{bmatrix},
\qquad
A^+ = \begin{bmatrix}
50 & 20 \\
20 & 9
\end{bmatrix},
\qquad
A^- = \begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}.
\end{equation*}

$\star$ %Define the above earlier?
Give an example?
%Make this a proposition?  State it later?
Give an interpretation for an arbitrary star network?

The coordinate ring of the variety of $2n \times 2n$ matrices is simply
$\cz$, where
$z$ is the $2n \times 2n$ matrix of indeterminates
$(z_{i,j})_{i,j \in [\ol n,n]}$.
Whereas $\Lambda_n(x,y)$ is a natural setting for generating functions
which record values of $\theta(D)$
for fixed $g \in \zbn$ and varying $\theta \in \trsp(\bn)$,
the ring $\cz$ is a natural setting for
generating functions
which record values of $\theta(D)$
for fixed $\theta \in \trsp(\bn)$ and varying $g \in \bn$.

%\ssec{Type-$C$ analogs of Lindstr\"om's Lemma}

\section{Old Type BC symmetric functions}

%\ssec{Type $B$ trace space}
%Analogous to $\trsp^A_n$ is the
%$\mathbb Z$-module
%of {\em $\bn$-traces},
%linear functionals $\theta: \bn \rightarrow \mathbb Z$ satisfying
%$\theta(gh) = \theta(hg)$ for all $g, h \in \bn$.
%By \cite[Thm.\,2.2.14]{HoefsmitThesis}, the $BC$-trace space $\btrsp_n$
%has rank equal to
%the number of ordered pairs $(\lambda,\mu)$ of partitions with
%$|\lambda| + |\mu| = n$.
%We call such a pair a {\em bipartition} of $n$
%and write $(\lambda,\mu) \vdash n$.
%(See also \cite[Thm.\,2.8]{HRam96}.)
%there is one irreducible character $\chi^{\lambda,\mu}$ of $\bn$
%for each ordered pair $(\lambda,\mu)$ of partitions with
%$|\lambda| + |\mu| = n$.  We will write $(\lambda,\mu) \vdash n$.
%Furthermore, these characters form a complete
%set of irreducibles.
%$\star$ Do we need to define the Hecke algebra
%of type $B_n$ over $\mathbb C(u,q^2)$ as Halverson and Ram~\cite{HRam96} do,
%or shall we define it over $\zqq$?
%\bp Do
%These span the trace space of $\bn$.
%Thus, the number of partition pairs of $n$ is the dimension of the
%$\mathbb Z[W$] trace space $\trsp(\bn)$.
%($\star$ Update the rest of this for type $BC$.)
%the number of integer partitions
%of $n$, the weakly decreasing positive integer sequences $\lambda  = (\lambda_1,\dotsc,\lambda_\ell)$ satisfying $\lambda_1 + \cdots + \lambda_\ell = n$.
%The $\ell = \ell(\lambda)$ components of $\lambda$ are called its {\em parts}
%and we let $|\lambda| = n$ 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
%  $\lambda_i^\tr = \# \{ j \,|\, \lambda_j \geq i \}$.

The bases of $\trsp(\bn)$ which correspond to the nonplethystic
symmetric function bases (\ref{eq:bcnpbases}) of $\Lambda_n(x,y)$
can be constructed from pairs of symmetric group trace bases, i.e.,
bases of
\begin{equation*}
\bigoplus_{m=0}^n \trsp(\mfs{m}) \otimes \trsp(\mfs{n-m}).
\end{equation*}
Specifically, given an $\mfs{m}$-character $\theta$,
we use the homomorphism $\varphi$ to define $\mfb{m}$-characters
%$1\theta$ and $\delta\theta$ by
\begin{equation*}
1\theta(w) \defeq \theta(\varphi(w)),
\qquad
\delta\theta(w) \defeq (-1)^{\ell_t(w)}\theta(\varphi(w)).
\end{equation*}
%Specifically, given an $\mfs{m}$-representation $\rho$ with character
%$\theta$, we use the homomorphism $\varphi$
%to define $\mfb{m}$-representations $1\rho$ and $\delta\rho$ with characters
%$1\theta$ and $\delta\theta$, respectively by
%  \begin{alignedat}{2}
%    1\rho(w) &\defeq \rho(\varphi(w)),
%    &\qquad 1\theta(w) &\defeq \theta(\varphi(w)),\\
%    \delta\rho(w) &\defeq (-1)^{\ell_t(w)}\rho(\varphi(w))
%    &\qquad \delta\theta(w) &\defeq (-1)^{\ell_t(w)}\theta(\varphi(w)).
%  \end{alignedat}
%Is that easier, or is it easier to say
%  \begin{alignedat}{2}
%    1\rho(s'_i) &= \rho(s_i), &\qquad \delta\rho(s'_i) &= \rho(s_i),\\
%    1\rho(t) &= I,           &\qquad \delta\rho(t) &= -I,
%  \end{alignedat}
%and that the characters are $1\theta$ and $\delta\theta$, respectively?
%($\star$ Keep both until it is clear which is better.)
Then given
a pair of bases
\begin{equation}\label{eq:basispair}
(\{ \theta^\lambda \,|\, \lambda \vdash 0,\dotsc,n \},
\{ \xi^\lambda \,|\, \lambda \vdash 0,\dotsc,n \})
\end{equation}
of $\trsp(\mfs0) \oplus \cdots \oplus \trsp(\mfs{n})$,
%families of symmetric group trace space bases
%  \Bigg( \bigcup_{m=0}^n \{ \theta^\lambda \,|\, \lambda \vdash m \},
%  \qquad
%  \bigcup_{m=0}^n \{ \xi^\mu \,|\, \mu \vdash m \} \Bigg),
we create a basis $\{ (\theta\xi)^{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$
of $\trsp(\bn)$
by defining traces
$q\theta_q$ of $\mfb{|\lambda|}$ and
$\delta\xi_q$ of $\mfb{|\mu|}$ and inducing
\begin{equation}\label{eq:inducedpair2}
(\theta\xi)^{\lambda,\mu} \defeq (q\theta^\lambda \otimes \delta\xi^\mu)
\upparrow_{\mfb{[1,|\lambda|]} \times \mfb{[|\lambda|+1,n]}}^{\bn}.
\end{equation}
%($\star$ Mention that for $\theta, \xi \in \{ \epsilon, \eta \}$,
%we can also write
%    (\theta\xi)^{\lambda,\mu} \defeq (1\theta^\lambda \otimes \delta\xi^\mu)
%  \upparrow_{\mfb{\lambda,\mu}}^{\bn},

These character bases
are related to one another just as their Frobenius images are:
\begin{equation*}
\begin{alignedat}{2}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
%    \sumsb{\alpha \vdash |\lambda|\\ \beta \vdash |\mu|}
K_{\alpha^{\tr}\ntnsp, \lambda}K_{\beta^{\,\tr}\ntnsp, \mu}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
K_{\alpha^{\tr}\ntnsp, \lambda}K_{\beta, \mu}
\phantom{\sum_{\mu \vdash n} K_{\mu,\lambda} s_\mu}   \\
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
K_{\alpha^{\tr}\ntnsp, \lambda}K_{\beta^{\,\tr}\ntnsp, \mu}
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
K_{\alpha^{\tr}\ntnsp, \lambda}K_{\beta, \mu}
\end{alignedat}
\end{equation*}
\begin{equation*}
\begin{alignedat}{2}
\phantom{= \sum_{\mu \vdash n} K_{\mu,\lambda} s_\mu}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
%    \sumsb{\alpha \vdash |\lambda|\\ \beta \vdash |\mu|}
K_{\alpha, \lambda}K_{\beta^{\,\tr}\ntnsp, \mu}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
K_{\alpha, \lambda}K_{\beta, \mu}
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
K_{\alpha, \lambda}K_{\beta^{\,\tr}\ntnsp, \mu}
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
K_{\alpha, \lambda}K_{\beta, \mu}
\end{alignedat}
\end{equation*}
where we may assume in all sums that $|\alpha| = |\lambda|$ and
$|\beta| = |\mu|$, since the Kostka numbers are $0$ otherwise.
(Compare to (\ref{eq:kostka}).)
%$\{K_{\lambda,\mu} \,|\, \lambda, \mu \vdash n \}$ are
%the {\em Kostka numbers}.

Five more bases of $\trsp(\bn)$ which do not consist of characters are
monomial-forgotten traces
%$\{ (\phi\phi)^{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$,
%$\{ (\phi\gamma)^{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$,
%$\{ (\gamma\phi)^{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$,
%$\{ (\gamma\gamma)^{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$,
and power sum traces
%$\{ (\psi\psi)^{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$,
which correspond to the
monomial-forgotten bases and nonplethystic power sum bases (\ref{eq:bcnpbases})
of $\Lambda_n(x,y)$ by

These non-character bases are related to irreducible characters
just as their Frobenius images are related to Schur functions:
\begin{equation*}
\begin{alignedat}{2}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
%    \sumsb{\alpha \vdash |\lambda|\\ \beta \vdash |\mu|}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
\phantom{\sum_{\mu \vdash n} K_{\mu,\lambda} s_\mu}   \\
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
\end{alignedat}
\end{equation*}
\begin{equation*}
\begin{alignedat}{2}
\phantom{= \sum_{\mu \vdash n} K_{\mu,\lambda} s_\mu}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
%    \sumsb{\alpha \vdash |\lambda|\\ \beta \vdash |\mu|}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
(f\ntnsp f)_{\lambda,\mu}
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
\end{alignedat}
\end{equation*}
\begin{equation*}
\begin{aligned}
&= \ntksp \sum_{(\alpha, \beta) \vdash n} \ntksp
\chi^\alpha(\lambda)\chi^\beta(\mu)
&= \ntksp \sum_{(\alpha,\beta) \vdash n} \ntksp
\chi^\alpha(\lambda)\chi^\beta(\mu)
\end{aligned}
\end{equation*}
where again we may assume in all sums that $|\alpha| = |\lambda|$ and
$|\beta| = |\mu|$.
(Compare to (\ref{eq:invkostka}).)

\section{Old Symmetric functions}

Three more bases of $\trsp(\sn)$ which do not consist of characters are
the power sum traces $\{ \psi^\lambda \,|\, \lambda \vdash n \}$,
monomial traces $\{ \phi^\lambda \,|\, \lambda \vdash n \}$,
and forgotten traces $\{ \gamma^\lambda \,|\, \lambda \vdash n \}$,
\begin{equation*}
\frobch(\psi^\lambda) = p_\lambda,
\qquad \frobch(\phi^\lambda) = m_\lambda,
\qquad \frobch(\gamma^\lambda) = f_\lambda.
\end{equation*}
These non-character bases are related to irreducible characters
just as their Frobenius images are related to Schur functions:
%which are related to the irreducible character bases by the same
%matrices of character evaluations and {\em inverse Kostka numbers} which
%relate power sum, monomial and forgotten symmetric functions to Schur functions,
\begin{equation}\label{eq:invkostka}
\begin{alignedat}{3}
\psi^\lambda &= \sum_{\mu \vdash n} \chi^\mu(\lambda) \chi^\mu,
&\qquad\qquad \phi^\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu} \chi^\mu,
&\qquad\qquad \gamma^\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu^{\tr}} \chi^\mu,
\\
p_\lambda &= \sum_{\mu \vdash n} \chi^\mu(\lambda) s_\mu,
&\qquad\qquad m_\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu} s_\mu,
&\qquad\qquad f_\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu^{\tr}} s_\mu,
\end{alignedat}
\end{equation}
where $\chi^\mu(\lambda) \defeq \chi^\mu(w)$ for any $w \in \sn$ having
$\ctype(w) = \lambda$,
and $\{ K^{-1}_{\lambda,\mu} \,|\, \lambda, \mu \vdash n \}$ are the {\em inverse
Kostka numbers}.
The power sum traces of $\trsp(\sn)$ also have the natural definition
\begin{equation*}
\psi^{\lambda}(w) = \begin{cases}
z_\lambda &\text{if $\ctype(w) = \lambda$},\\
0 &\text{otherwise},
\end{cases}
\end{equation*}
where $z_\lambda$ is the product
$\lambda_1 \cdots \lambda_\ell \alpha_1! \cdots \alpha_n!$,
and $\alpha_i$ is the number of parts of $\lambda$ equal to $i$.

Three more bases of $\trsp(\hnq)$ which do not consist of characters are
the power sum traces $\{ \psi_q^\lambda \,|\, \lambda \vdash n \}$,
monomial traces $\{ \phi_q^\lambda \,|\, \lambda \vdash n \}$,
and forgotten traces $\{ \gamma_q^\lambda \,|\, \lambda \vdash n \}$,
%  \frobch(\psi^\lambda) = p_\lambda,
%  \qquad \frobch(\phi^\lambda) = m_\lambda,
%  \qquad \frobch(\gamma^\lambda) = f_\lambda.
%These non-character bases are related to irreducible characters
%just as their Frobenius images are related to Schur functions:
%which are related to the irreducible character bases by the same
%matrices of character evaluations and {\em inverse Kostka numbers} which
%relate power sum, monomial and forgotten symmetric functions to Schur functions,
\begin{equation}\label{eq:invkostka}
\psi_q^\lambda = \sum_{\mu \vdash n} \chi^\mu(\lambda) \chi_q^\mu,
\qquad\quad
\phi_q^\lambda = \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu} \chi_q^\mu,
\qquad\quad
\gamma_q^\lambda = \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu^{\tr}} \chi_q^\mu,
%  \begin{alignedat}{3}
%    \psi^\lambda &= \sum_{\mu \vdash n} \chi^\mu(\lambda) \chi^\mu,
%    &\qquad\qquad \phi^\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu} \chi^\mu,
%    &\qquad\qquad \gamma^\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu^{\tr}} \chi^\mu,
%   \\
%    p_\lambda &= \sum_{\mu \vdash n} \chi^\mu(\lambda) s_\mu,
%    &\qquad\qquad m_\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu} s_\mu,
%    &\qquad\qquad f_\lambda &= \sum_{\mu \vdash n} K^{-1}_{\lambda,\mu^{\tr}} s_\mu,
%  \end{alignedat}
\end{equation}
where $\chi^\mu(\lambda) \defeq \chi^\mu(w)$
for any $w \in \sn$ having cycle type $\lambda$,
and $\{ K^{-1}_{\lambda,\mu} \,|\, \lambda, \mu \vdash n \}$
are the {\em inverse Kostka numbers}.
The power sum traces of $\trsp(\sn)$ also have the natural definition
\begin{equation*}
\psi^{\lambda}(w) = \begin{cases}
z_\lambda &\text{if $\ctype(w) = \lambda$},\\
0 &\text{otherwise},
\end{cases}
\end{equation*}
where $z_\lambda$ is the product
$\lambda_1 \cdots \lambda_\ell \alpha_1! \cdots \alpha_n!$,
and $\alpha_i$ is the number of parts of $\lambda$ equal to $i$.

\ssec{Hecke algebra to symmetric group maps}

A $q$-analog $\varphi_q: \hbnq \rightarrow \hnq$ of
the homomorphism $\varphi:W \rightarrow \sn$ is
$(T_{s'_i}, T_t) \mapsto (T_{s'_i}, q)$
for all $i \geq 1$.
A $q$-analog $\varphi_{q}^-: \hbnq \rightarrow \hnq$ of the map $\varphi^-$ is
$(T_{s'_i}, T_t) \mapsto (T_{s'_i}, -1)$
for all $i \geq 1$.
A $(q,Q)$-analog $\varphi_{q,Q}: \hbnQ \rightarrow \hnq$ of
the homomorphism $\varphi:W \rightarrow \sn$ is
$(T_{s'_i}, T_t) \mapsto (T_{s'_i}, Q)$
for all $i \geq 1$.
Again, a $(q,Q)$-analog $\varphi_{q,Q}^-: \hbnQ \rightarrow \hnq$ of
the map $\varphi^-$ is
$(T_{s'_i}, T_t) \mapsto (T_{s'_i}, -1)$
for all $i \geq 1$.

\section{dunno}

($\star$ Introduce this.)

($\star$ Do this in greater generality: look at sequences
$(\inc(P_{I_1}), \dotsc, \inc(P_{I_\ell}))$ of type $\lambda$ or $\mu$
and acyclic orientations of each that have only one source.
Make sure grounded vertices are in the left part.)

($\star$ Given $n$-element marked type-$\msfBC$ indifference graph $G$
and $I \subseteq [n]$,
define $\mathcal O(G, I)$ to be the set of pairs
where $O_1$ is a marked acyclic orientation of $G_I$ containing
all grounded vertices of $G$ and having one source,
and $O_2$ is an acyclic orientation of $G_{[n]\ssm I}$ having one source.
Or define something consistent with the equation below.)
\begin{thm}\label{t:psiorient}
Let $w \in \bn$ be $\msfBC$-smooth and fix $k \leq n$.  Then we have
\begin{equation*}
(\psi\psi)^{k,n-k}(\btc w1) =
\sumsb{I \subseteq [n]\\|I|=k}
%\{ (O_1, O_2) \in
|\mathcal O(\inc(P_I),\inc(P_{[n]\ssm I}))|.
\end{equation*}
\end{thm}
\begin{proof}
$\star$ Do this.  Also, change all $F_w$-tableaux of type $e$ to $P$-tableaux.
\end{proof}

($\star$ Mention that this can be extended to a combinatorial
interpretation of $(\psi\psi)^{\lambda,\mu}$ orientations of graphs.  Cite

\section{Maybe save}
for $\theta_1, \theta_2 \in \{\eta, \epsilon \}$ and
$(\lambda,\mu) \vdash n$
with $|\lambda| = m$ that
\begin{equation}\label{eq:wrthimm}
\bnimm{(\theta_1\theta_2)^{\lambda,\mu}}(z) =
\sumsb{I\subseteq[n]\\|I|=m}\simm{m}{\theta_1^\lambda}(z^+_{I,I})\simm{n-m}{\theta_2^\mu}(z^-_{[n]\ssm I,[n]\ssm I}).
\end{equation}
%where the sum is over all $m$-element sets of $[n]$.

$\star$ Can we just write
\begin{equation*}
\bnimm{(\theta\xi)^{\lambda,\mu}}(z) =
\sum_I\simm a{\theta^\lambda}(z^+_{I,I})\simm{n-a}{\xi^\mu}(z^-_{[n]\ssm I,[n]\ssm I})
\end{equation*}
for all $\bn$-characters of the form $(\theta\xi)^{\lambda,\mu}$
defined in (\ref{eq:inducedpair})?  Yes:

\section{Introduction 2}\label{s:intro2}

Mention chromatic symmetric functions and how they serve as generating
functions for Kazhdan--Lusztig basis elements of type $A$. Mention pattern
avoidance and refer reader to that section.

Pose problem of doing this in type $BC$.

Define Kazhdan--Lusztig basis elements here, at least for the smooth ones
(all coefficients equal to $1$).

\section{Old Put This Somewhere}

\ssec{Path families of type $\msfBC$}\label{s:pathfam}

Let $G_{[a,b]}$ be a type-$\msfBC$ simple star network, and let
$\pi = (\pi_{\ol n}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_n )$
be a sequence of paths covering all edges of $G_{[a,b]}$.
If there is an element $w = w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n \in \bn$
such that for all $i$, $\pi_i$ is a path from source $i$ to sink $w_i$,
then call $\pi$ a
{\em type-$\msfBC$ path family of type $w$ which covers $G_{[a,b]}$}.
($\star$ Rethink name.)
Observe that the condition $w_{\ol i} = \ol{w_i}$ on elements of $\bn$
implies a vertical mirror symmetry of $\pi$.
Now let
$\hat G = G_{[a_1,b_1]} \bullet \cdots \bullet G_{[a_r,b_r]}$,
be a general type-$\msfBC$ star network, and let
$\pi = (\pi_{\ol n}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_n )$
be a sequence of paths covering all edges of $\hat G$
such that for all $i$, $\pi_i$ is a path from source $i$ to sink $w_i$.
If for $i = 1,\dotsc,r$, there exist elements $v_{(i)} \in \bn$ such that
the restriction of $\pi$ to $G_{[a_i, b_i]}$ is
a type-$\msfBC$ path family of type $v^{(i)}$ which covers $G_{[a_i,b_i]}$,
then say $\pi$ is a
{\em type-$\msfBC$ path family of type $w$ which covers $G_{[a,b]}$}.

%Call a path $\pi_i$ in a path family {\em grounded}
%if it intersects $\pi_{\ol i}$.
%Call $\pi$ a
%$\hat G = G_{[a_1,b_1]} \bullet \cdots \bullet G_{[a_r,b_r]}$,
Again, $\pi$ will have mirror symmetry about the horizontal line.

%$\star$ Define type of a path family.
$\star$ Give examples.

%\section{Put this somewhere$}
%And put the next theorem later, in another section?
%Or replace this with the next theorem?
\begin{cor}\label{c:bckl}
Let $w \in \bn$ \avoidp. Then the type-$\msfBC$ Kazhdan--Lusztig basis element
\begin{equation}\label{eq:bckl}
\btc wq = \sum_\pi T_{\type(\pi)},
\end{equation}
where the sum is over all $\bn$ path families
%of type $v \in \bn$
covering $G_w$.
\end{cor}
\begin{proof}
By Theorem~\ref{t:zigzagbc}, the right-hand-side of (\ref{eq:bckl}) is
\begin{equation*}
%C_w^{\bn}(q) =
\sum_{v \leq_{\bn} w} T_v.
\end{equation*}
By Proposition~\ref{p:abcsmoothkl},
%(Lakshmibai-Sandhya/Seshadri? result),
this is $\btc wq$.
%  By Proposition~\ref{p:abcbruhat}
%  we have $v \leq_{\bn} w$ if and only if $v \leq_{\snn} w$.
%  By ?, $Z(w)$ is a zig-zag network, and
%  by \cite[Lem.\,5.3]{SkanNNDCB}
%  there exists exactly one path family of type $v$ covering $Z(w)$
%  if $v \leq_{\snn} w$ and no such path family otherwise.
%  Combine Theorems~\ref{t:bcsmoothkl} and \ref{t:zigzagbc}.
%  Since
%  \begin{equation*}
%    C_w^{\bn}(1) = \sum_{v \leq_{\bn} w} v,
%  \end{equation*}
%(state that result somewhere?) the result follows immediately
%from Theorem~\ref{t:zigzagbc}.
\end{proof}

\ssec{Defect formula}

Say that a star network {\em represents} an element
$g = \sum_{w \in \bn} a_w T_w \in \hbnq$
if for every $w \in \bn$ we have
\begin{equation*}
a_w = \sum_\pi q^{\dfct(\pi)},
\end{equation*}
where the sum is over all type-$\msfBC$ path families of type $w$.
(See Subsection~\ref{s:pathfam}
and Billey-Warrington~[11],
for definitions.)
%where $a_w$ is the

For each reversal
$\{ s'_{[\ol a, a]} \,|\, 0<a \leq n \} \cup \{ s'_{[a,b]} \,|\, 0<a<b \leq n \}$,
the corresponding simple star network $G_{[\ol a, a]}$ ($G_{[a,b]}$)
represents the Kazhdan--Lusztig basis element $\wtc{s_{[\ol a, a]}}q$

%\bp Show that for each subinterval $[a,b]$ of $[0,n-1]$,
%the parabolic subgroup of $\bn$ generated by
%generators $\{ s_a,\dotsc,s_b \}$
%%of $\bn$ generate a
%of $\bn$ having maximal element $w$ do we
%have a simple star network representing the Kazhdan--Lusztig basis element
%$\wtc wq$?
%%How do we compute from the star network the coefficient of
%%$T_w$ in $\wtc wq$?

%\bp Is there a defect formula (a la Billey-Warrington~[11],
%Clearwater-Skandera~[15], and  Deodhar~[20])
%that allows one to represent products of the Kazhdan--Lusztig basis
%elements $\wtc wq$ above as star networks
%(concatenations of simple star networks)?

%\bp What is the relationship between $\hbnq$ and $H_{2n}(q)$?

\bp
%Is there a nice characterization of the elements $w \in \bn$ for
Show that the star network for $\btc wq$ is a zig-zag network
if and only if $w$ \avoidsp.
\ep

\ssec{$2n \times 2n$ matrices}

\ssec{Type $BC$ KL basis element factorization}

\bp Show that when the long one-line notation of $w \in \bn$ \avoidsp,
there exists a factorization of $w$ as a product
$s'_{[a_1,b_1]} \cdots s'_{[a_r, b_r]}$ of type-$\msfBC$ reversals such that we also have
\begin{equation*}
\btc wq = f(\qp12) \btc{s'_{[a_1,b_1]}}q \cdots \btc{s'_{[a_r,b_r]}}q,
\end{equation*}
where $f$ is some rational function.
\ep

\section{Old BC unit interval order bijection and material after}

%($\star$ Change boxes above to circles.)
%are placed in a box.
%We call the elements placed in boxes {\em grounded elements}.
%($\star$ Mention grounded paths earlier and use these to motivate the definition
%of grounded elements.)
\begin{prop}\label{p:rtorw}
The map $F_w \mapsto Q(w)$
%$w \mapsto Q(w)$
$\dnet{BC}{[\ol n,n]}$
%  $\{ w \in \bn \,|\, F_w \in \dnet{BC}{[\ol n,n]} \}$
%  ($\star$ or elements of $\bn$ avoiding the signed patterns $312$, $3\ol12$,
%  $\ol2\ol1$, $\ol21$)
%$\dnet{BC}{[\ol n,n]}$
to type-$\msfBC$ unit interval orders is bijective.
This bijection
%Its description above
is equivalent to the following algorithm.
\begin{enumerate}
\item Let $j$ be the greatest index in $[\ol n,\ol 1]$
satisfying $w^{-1}_j < 0$ and define
$w'_{\ol n} \cdots w'_{\ol 1} w'_1 \cdots w'_n$ by
\begin{enumerate}
\item $w'_i = w^{-1}_i$ for $i = \ol n, \dotsc, j$,
\item $w'_i = |i|$ for $i = j+1 \dotsc, \ol 1$,
\item $w'_i = w_i$ for $i = 1,\dotsc,n$.
\end{enumerate}
\item Define $m_{\ol n} \cdots m_{\ol 1} m_1 \cdots m_n$ by
$m_i = \max\{w'_{\ol n}, \dotsc, w'_i \}$.
\item For $i = 1,\dotsc, n-1$, define $i <_{Q(w)} m_i+1, \dotsc, n$.
\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{prop}
\begin{proof}
To see that the map is injective, consider
$F_v \neq F_w$ in $\dnet{BC}{[\ol n,n]}$.
%\subset \znet{BC}{[\ol n,n]}$.
% and let $(\pi_{\ol n}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_n)$,
% $(\sigma_{\ol n}, \dotsc, \sigma_{\ol 1}, \sigma_1, \dotsc, \sigma_n)$
% be the unique path families of type $e$ covering these, respectively.
By \cite[Thm.\,4.4]{CHSSkanEKL} we have $P(v) \neq P(w)$,
since for each fixed unit interval order $P$ on $2n$ elements,
the set
$\{ F_w \in \znet{BC}{[\ol n,n]} \,|\, P(w) = P \}$
contains exactly one type-$\msfBC$
descending star network: the rearrangement
$F'_{[a_{u_1},b_{u_1}]} \bullet \cdots \bullet F'_{[a_{u_t},b_{u_t}]}$ of
(\ref{eq:bcbulletconcat}) satisfying $a_{u_1} > \cdots > a_{u_t}$
(and $b_{u_1} > \cdots > b_{u_t}$).
Now let $P'(v)$, $P'(w)$ be the subposets of $P(v)$ and $P(w)$ induced
by elements $\{ 1, \dotsc, n \}$.  If $P'(v) \neq P'(w)$ then
we clearly have $Q(v) \neq Q(w)$.  Suppose
therefore that $P'(v) = P'(w)$.  Since $P(v) \neq P(w)$,
there must be two indices $i \neq j$ such that
elements $1, \dotsc, i$ of $P'(v)$ are grounded, and elements
$1, \dotsc, j$ of $P'(w)$ are grounded.
Again we have $Q(v) \neq Q(w)$.

To see that the map is surjective, consider a type-$\msfBC$ unit interval
order $Q$ on $n$ elements with elements labeled
as in (\ref{eq:altitude}) -- (\ref{eq:altrespect})
and with a subset $\{1, \dotsc, i\}$ of minimal elements circled,
for some $i$.
%Temporarily ignoring the $i$ circles, let
Let $F_u \in \dnet{A}{[n]}$, $u \in \sn$,
be the descending star network corresponding
to $Q$ viewed as an ordinary poset, ignoring circles,
and write $F_u = F_{[a_1,b_1]} \bullet \cdots \bullet F_{[a_t,b_t]}$ as
in Definition~\ref{d:adsn}.  Now construct
$F'_{[a_1,b_1]} \bullet \cdots \bullet F'_{[a_t,b_t]}
\bullet F'_{[\ol i, i]}$ in $\dnet{BC}{[\ol n,n]}$ and call this $F_w$
for $w \in \bn$.
It is easy to see that we have $F_w \mapsto Q$,
%This network maps to the decorated poset $Q$
%and has the form $F_w$ for some $w \in \bn$.
%and thus $w$
%  F_w =
%  This network
i.e., $Q = Q(w)$.

($\star$ Redo this completely.)
To justify the second statement, observe first that since
$w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n$ consists of distinct letters
we have $m_i \geq i$ for all $i$.
Now fix $i > 0$,
let $\sigma$ be the unique path family of type $e$ covering $F_w$,
and let
%Since there exists a path family
%= (\pi_{\ol n}, \dotsc, \pi_{\ol1}, \pi_1,\dotsc,\pi_n)$
be the unique path family of type $w$ covering $F_w$.
At each vertex where two paths of $\sigma$ intersect
(necessarily without crossing),
two corresponding paths of $\pi$ intersect and cross.

%The family $\pi$ is formed from $\sigma$ by
By the definition of $m_i$, we have some index $h \leq i$ satisfying
$w_h = m_i$.
Thus the path $\pi_h$ starts at source $h < i$
and terminates at sink $m_i \geq i$, crossing the path $\sigma_i$.
It follows that there exists a path in $F_w$ from source $i$ to sink
$m_i \geq i$, and by
%This path necessarily
\cite[Lem.\,3.5]{CHSSkanEKL}
that paths $\sigma_i$, $\sigma_{m_i}$ intersect
or are equal if $i = m_i$.
Thus in the poset $P(w)$, elements $i$ and $m_i$ are incomparable or equal.
%to $m_i$ in $P(w)$.
Now consider the path $\sigma_{m_i + 1}$ and assume that
it intersects path $\sigma_i$.  Then there is a path in $F_w$ from
source $i$ to sink $m_i+1$ and by
\cite[Obs.\,3.4]{CHSSkanEKL},
the intervals
and indices $i$, $m_i+1$ satisfy one of the conditions
\begin{enumerate}
\item one interval $[c_k,d_k]$ contains both $1$ and $m_i+1$,
%  $1$ and $m_i+1$ belong to a common interval
\item two intervals $[c_k,d_k] \prec [c_\ell,d_\ell]$ contain $1$ and $m_i+1$,
\end{enumerate}
In the first case,
%the interval can not satisfy $c_k > 1$, because
the intersection of $\sigma_i$, $\sigma_{m_i+1}$ in $F_{[c_k,d_k]}$ implies
that $\pi_i$, $\pi_{m_i+1}$ cross in $F_{[c_k,d_k]}$,
and that for some $h \leq i$ the path $\pi_h$ crosses both and terminates
at sink $d_k \geq m_i+1$.  This forces $w_h \geq m_i+1$, contradicting
that $w_h \leq m_i$.
%$\max\{w_{\ol n}, \dotsc, w_i \} = m_i$.
%Suppose therefore that the interval has the form $[\ol a, a]$.
In the second case, while there is a path in $F_w$ from source $i$ to sink
$m_i+1$, the paths $\sigma_i$, $\sigma_{m_i+1}$ do not cross.
Thus we have $i <_{P(w)} m_i+1$.
%in the poset $P(w)$, elements $i$ and $m_i$ are incomparable or equal.
Since no path $\sigma_{m_i+2},\dotsc,\sigma_n$ crosses $\sigma_{m_i+1}$
we also have $i <_{P(w)} m_i+2,\dotsc,n$.
\cite[Lem.\,3.5]{CHSSkanEKL} implies that
the condition $w_i = \ol a < 0$ for $i > 0$
(and necessarily $w_{\ol i} = a > 0$) is equivalent
to $\sigma_a$ intersecting $\sigma_{\ol a}$, and therefore to $a$ being
a circled element of $Q(w)$.
($\star$ Review the second part of this proof and try to shorten it and/or make
it easier to understand.
Maybe state the cited Observation 3.4 and Lemma 3.5 in this paper?)
\end{proof}

Given a type-$\msfBC$ unit interval order $Q$, define its {\em incomparability
graph} $\inc(Q)$ to be the decorated graph
whose vertices are the elements of $Q$, maintaining circles,
and whose edges are the pairs of incomparable elements of $Q$.
For $w \in \bn$ \avoidingp, write $G(w)$ for $\inc(Q(w))$
and call such a decorated graph a {\em type-$\msfBC$ unit interval graph}.
It is possible for nonisomorphic type-$\msfBC$ unit interval orders
to have isomorphic incomparability graphs.
For instance, the
%$Q$, define its {\em incomparability
thirteen type-$\msfBC$ unit interval graphs on three elements are
\begin{equation}\label{eq:uig}
\raisebox{-4mm}{
\includegraphics[height=10mm]{uig1.eps}}
\raisebox{-2mm},
\end{equation}
with the third graph here being the incomparability graph of the
third and fourth type-$\msfBC$ unit interval orders in (\ref{eq:uioc}).
($\star$ Redo this in tikz with labeled vertices.)

%($\star$ Define type-$\msfBC$ posets and their incomparability graphs in
%greater generality? Define type-$\msfBC$ graphs in greater generality?)
%($\star$ List the graphs corresponding to the posets above.)
%($\star$ Keep this somewhere?)
%For each path family $\pi$ of type $v \in \bn$ covering $F_w$,
%the poset $Q(\pi)$ defined in Subsection~\ref{ss:Aposetgraph}
%has induced subposets $Q(\pi)_{[n]}$ and $Q(\pi)_{[\ol n, \ol 1]}$ which are dual to
%one another.
%($\star$ Fix this; it is repeated later?)

It is easy to see that chains in $Q(w)$ correspond to independent sets
(edge-free induced subgraphs) of $G(w)$, and that each such chain
or independent set contains at most one grounded element.
Define a {\em $\msfBC$-coloring} of a type-$\msfBC$ unit interval graph
$G = (V,E)$ to be a function $\kappa: V \rightarrow \mathbb Z \ssm \{0\}$
with the property that for each grounded vertex $v \in V$ we have
$\kappa(v) > 0$. Call the coloring {\em proper} if $(a,b) \in E$ implies
that $\kappa(a) \neq \kappa(b)$.
It is easy to see that a sequence $(C_{\ol k}, \dotsc, C_{\ol 1}, C_1, \dotsc, C_m)$
of pairwise disjoint chains of cardinalities
$\mu_k, \dotsc, \mu_1, \lambda_1, \dotsc, \lambda_m$
in which all grounded elements of $P$ belong to chains $C_1,\dotsc,C_m$
%$\alpha_1,\dotsc, \alpha_r$
corresponds to a proper coloring of $G$ in which
\begin{enumerate}
\item $\mu_i$ vertices have color $\ol i$, for $i=1,\dotsc, k$,
\item $\lambda_i$ vertices have color $i$, for $i=1,\dotsc, m$.
%\item all grounded vertices $v$ satisfy $\kappa(v) > 0$.
\end{enumerate}
Call a coloring which satisfies these conditions a
{\em proper $\msfBC$-coloring}.
Define an acyclic orientation of a type-$\msfBC$ unit interval graph
to be a directed graph on the same verices, with the same circles,
in which each undirected edge $\{ a, b \}$ is replaced with one of
the directed edges $(a,b)$ or $(b,a)$.
%($\star$ Redefine {\em proper} to include these conditions?)

%($\star$ Finish this and draw posets.)
%$\star$ Mention that each type-$\msfBC$-path family forms another poset.
%(This poset is not a type-$\msfBC$ unit interval order,
%but it is a type-$\msfBC$ poset.)
%The $2n$ paths of $\pi$ naturally form a poset which we call $Q(\pi)$, defined
%by $\pi_i <_{Q(\pi)} \pi_j$ if $\pi_i$ and $\pi_j$ share no vertex and $i < j$.
%which we call $Z$-tableaux.
%To combinatorially interpret evaluations of $\bn$-traces, we will
%$\star$ Do an example
%\ssec{Incomparability graphs}
%Given graph $G = (V, E)$ call a directed graph $O = (V, E')$
%an {\em orientation} of $G$ if $O$ is obtained from $G$
%by replacing each undirected edge $\{i, j\} \in E$
%with exactly one of the directed edges $(i, j)$ or $(j, i)$.
%Call $O$ {\em acyclic} if it has no directed cycles.

($\star$ Is there anything special that needs to be said about
orientations and acyclicity in type $\msfBC$? Define a marked acyclic
orientation of $\inc(Q)$.)

($\star$ What pairs of sequences $(v_1 \cdots v_k, v_{k+1} \cdots v_n)$
correspond to what pairs of acyclic orientations of subgraphs of $Q(w)$?
In type $\msfA$, acyclic orientations of $G(w)$
correspond to
sequences $(v_h, \dotsc, v_l)$ of elements of $P(w)$
satisfying $v_i \not \geq_{P(w)} v_{i+1}$ for $i=h,\dotsc,l-1$.)

($\star$ Have Gavin look for type-$\msfBC$ analogs of the type-$\msfA$
evaluations of $\psi_q^\lambda$.)
%of elements of $P(w)$
%satisfying $v_i \not \geq_{P(w)} v_{i+1}$ for $i=1,\dotsc,n-1$ correspond
%to acyclic orientations of $G(w)$. ($\star$ Cite this.)

($\star$ Have Gavin look at Zaslavsky--Harary signed graphs and their
colorings?  Do some subset of signed graphs correspond to incomparability
graphs of type-$\msfBC$ unit interval orders?  Do colorings of the subset
of signed graphs correspond to colorings of incomparability graphs
of type-$\msfBC$ unit interval orders?)

%($\star$ Have Gavin

\section{Old type-$\msfA$ Hessenberg varieties}

\ssec{Hessenberg varieties}

Define a {\em complete $n$-dimensional flag over $\mathbb C$}
to be a sequence $F = (V_0, V_1, \dotsc, V_n)$
of subspaces with
$0 = V_0 \subset V_1 \subset \cdots \subset V_n = \mathbb C^n$
and $\dim V_i = i$ for all $i$.
Let $\mathcal F_n$ denote the variety of all complete $n$-dimensional flags
over $\mathbb C$.

Given a unit interval order $P$ and its corresponding
$312$-avoiding permutation $w$, define
the word $m_1 \cdots m_n$
by $m_i = \max \{ w_1,\dotsc, w_i \}$ and call this the
{\em Hessenberg word} of $w$.
Such words are characterized by 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}
For (any) fixed diagonal matrix
$D \in \mathrm{GL}_n(\mathbb C)$ define the
{\em regular semisimple type-$\msfA$
Hessenberg variety associated to $w$} (or $P$, or $m_1 \cdots m_n$)
to be the subvariety
\begin{equation*}
\mathcal H(w) =
%  \mathcal H(P) =
%  \mathcal H(m_1 \cdots m_n) =
\{ F \in \mathcal F_n \,|\,
DV_i \subseteq V_{m_i} \text{ for all } 1 \leq i \leq n-1 \}.
\end{equation*}

The cohomology of $H(w)$ vanishes in odd degree,
\begin{equation*}
H^*(\mathcal H(w)) = \bigoplus_{j=0}^{\ell(w_0w)} H^{2j}(\mathcal H(w)),
\end{equation*}
with each graded component having the structure of an $\sn$-module
via an action defined by Tymoczko~[66], [67].
Let $\mathrm{ch}(H^{2j}(\mathcal H(w)))$ be
the Frobenius characteristic of the character
of the module $H^{2j}(\mathcal H(w))$.
%($\star$ Explain that the notation
Shareshian and Wachs conjectured~\cite[Conj.\,10.1]{SWachsChromQF}
that if we expand the 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]$
as a polynomial in $q$ with coefficients
%are symmetric functions
in $\Lambda_n$, we obtain a generating function for the above
Frobenius characteristics.
This was proved by Brosnan--Chow~[13]
and Guay-Paquet~[28].
Combining this fact 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 $312$ and let $P$ be its corresponding
unit interval order.
%  and let $e$ be the number of edges in $\inc(P)$.
We have
\begin{equation}\label{eq:tymXY}
\sum_{j=0}^{\ell(w_0w)} \ntksp \mathrm{ch}(H^{2j}(\mathcal H(w)))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
$H^{2j}(\mathcal H(w))$
belongs to $\spn_{\mathbb N}\{ \chi^\lambda \,|\, \lambda \vdash n \}$.
Thus by (\ref{eq:frobchs})
its Frobenius characteristic
%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 $w \in \sn$ avoiding the pattern $312$ and $j = 0,\dotsc,\ell(w_0w)$,
the $\sn$-module $H^{2j}(\mathcal H(w))$ 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
Algorithm~\ref{a:ptow}, 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 all $w \in \sn$ and all $\lambda \vdash n$,
we have $\phi_q^\lambda(\wtc wq) \in \mathbb N[q]$.
\end{enumerate}
\end{prob}
The three statements in the problem satisfy the implications
$(1) \Rightarrow (2) \Leftarrow (3).$
For recent progress on these problems, see, e.g., ($\star$ Cite some paper.)

\section{A unit interval order on 6 elements}

\begin{tikzpicture}[scale=.4,baseline=0]%s2s22
\draw[fill] (0,2.5) circle [radius=0.15];
\draw[fill] (1,1.5) circle [radius=0.15];
\draw[fill] (0,.5) circle [radius=0.15];
\draw[fill] (0,-.5) circle [radius=0.15];
\draw[fill] (1,-1.5) circle [radius=0.15];
\draw[fill] (0,-2.5) circle [radius=0.15];
\draw[-,thick] (0,-.5) to [out=120,in=240] (0,2.5);
\draw[-,thick] (0,-2.5) to [out=120,in=240] (0,.5);
\draw[-,thick] (0,-2.5) -- (0,-.5);
\draw[-,thick] (0,2.5) -- (0,.5);
\draw[-,thick] (0,-2.5) -- (1,1.5);
\draw[-,thick] (0,2.5) -- (1,-1.5);
\node at (-.9,2.5) {$\scriptstyle 3$};
\node at (1.5,1.5) {$\scriptstyle 2$};
\node at (-.9,0.5) {$\scriptstyle 1$};
\node at (-.9,-0.5) {$\scriptstyle \ol 1$};
\node at (1.5,-1.5) {$\scriptstyle \ol 2$};
\node at (-.9,-2.5) {$\scriptstyle \ol 3$};
\end{tikzpicture}\, ,

\section{Old bijection proof about unit interval orders}

To see that the map is injective, consider
$F_v \neq F_w$ in $\dnet{BC}{[\ol n,n]} \subset \znet{BC}{[\ol n,n]}$.
and let $(\pi_{\ol n}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_n)$,
$(\sigma_{\ol n}, \dotsc, \sigma_{\ol 1}, \sigma_1, \dotsc, \sigma_n)$
be the unique path families of type $e$ covering these, respectively.
By \cite[Thm.\,4.4]{CHSSkanEKL} we have that $P(v) \not \cong P(w)$,
since for each unit interval order $P$ on $2n$ elements,
the set
$\{ F_w \in \znet{BC}{[\ol n,n]} \,|\, P(w) = P \}$
contains exactly one type-$\msfBC$
descending star network: the rearrangement
$F'_{[a_{u_1},b_{u_1}]} \bullet \cdots \bullet F'_{[a_{u_t},b_{u_t}]}$ of
(\ref{eq:bcbulletconcat}) satisfying $a_{u_1} > \cdots > a_{u_t}$
(and $b_{u_1} > \cdots > b_{u_t}$).
Now let $P'(v)$, $P'(w)$ be the subposets of $P(v)$ and $P(w)$ induced
by elements $\{ 1, \dotsc, n \}$.  If $P'(v) \not \cong P'(w)$ then
clearly we have $Q(v) \not \cong Q(w)$.  Suppose
therefore that $P'(v) \cong P'(w)$.  Since $P(v) \not \cong P(w)$,
there must be two indices $i \neq j$ such that
elements $\pi_1, \dotsc, \pi_i$ of $P'(v)$ are grounded, and elements
$\sigma_1, \dotsc, \sigma_j$ of $P'(w)$ are grounded.
Again we have $Q(v) \not \cong Q(w)$.

%  Let $\hat Q$ denote its dual, and combine $Q$, $\hat Q$
%  by declaring nongrounded elements of $\hat Q$ to be less than
%  all elements of $Q$, and grounded elements of $\hat Q$ to
%  be less than nongrounded elements of $Q$.  By the bijection
%  following (\ref{eq:posetex}), the resulting poset is $P(w)$ for some
%  $w \in \mfs{[\ol n,n]}$.
%  in
%  To justify the bijection, let $\mathcal P_{2n}$ be
%  %  Define
%  the subset of $2n$-element type-$\msfA$ unit interval orders
%  which are path posets of type-$\msfBC$ zig-zag networks,
%%  $\znet{BC}{[\ol n,n]}$,
\begin{equation*}
\mathcal P_{2n} \defeq \{ P(w) \,|\,
%w \in \mfs{[\ol n,n]},
F_w \in \znet{BC}{[\ol n,n]} \},
\end{equation*}
and let $\mathcal Q_n$ be the set
of all $n$-element type-$\msfBC$ unit interval orders.
%  By \cite[Thm.\,4.1, Thm.\,4.4]{CHSSkanEKL}, the
%  map $w \mapsto P(w)$ from $\bn$ to $$2n$-element unit interval orders
%  is surjective.
%Thus the map $w \mapsto P(w)$ is surjective.
%Furthermore, it
%  It is easy to see that for each
By \cite[Thm.\,4.4]{CHSSkanEKL}, we have that for
unit interval order $P \in \mathcal P_{2n}$, the set
$\{ F_w \in \znet{BC}{[\ol n,n]} \,|\, P(w) = P \}$
contains exactly one type-$\msfBC$
descending star network: the rearrangement
$F'_{[a_{u_1},b_{u_1}]} \bullet \cdots \bullet F'_{[a_{u_t},b_{u_t}]}$ of
(\ref{eq:bcbulletconcat}) satisfying $a_{u_1} > \cdots > a_{u_t}$
(and $b_{u_1} > \cdots > b_{u_t}$.)
Thus we have
\begin{equation*}
\mathcal P_{2n} = \{ P(w) \,|\, w \in \bn, F_w \in \dnet{BC}{[\ol n,n]} \}.
\end{equation*}
It is easy to see that the map from $\mathcal P_{2n}$ to $\mathcal Q_n$
(before (\ref{eq:uiocc})) ($\star$ Fix this reference.)
is invertible: given $Q(w) \in \mathcal Q_n$,
\begin{enumerate}
\item let $\hat Q(w)$ denote its dual,
%    \item form the disjoint union of $Q(w)$ and $\hat Q(w)$,
\item combine $Q(w)$ and $\hat Q(w)$ by declaring
nongrounded elements of $\hat Q(w)$ to be less than
all elements of $Q(w)$ and grounded elements of $\hat Q(w)$ to
be less than nongrounded elements of $Q(w)$.
\end{enumerate}
Composing this with the bijection $w \mapsto F_w$
following (\ref{eq:posetex}),
we have the first statement of the proposition.

\section{An old proof}

and is $0$ otherwise.  This corresponds to
$Q$ being a chain with

which do not

is the same, but is $0$ if there are any grounded elements.
Then by (bijection result again),
path families (\ref{eq:2npathfamily}) correspond bijectively to
descent-free $Q$-permutations and to excedance-free $Q$-permutations,
with no grounded paths.  ($\star$ Mention how these correspond
to certain acyclic orientations as well.)

Now rewrite $\det(A^+)$ and $\det(A^-)$ to match the above.  Use material
from below?

($\star$ Probably remove most of what is below, after rewriting it above.)

%Let $\sigma$ be the unique path family of type $e$ covering $Z_{I,I}$.
All path families
%$\pi = (\pi_{\ol k}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_k)$
covering $Z_{I\cup \ol I,I \cup \ol I}$ which are counted by $\perm(B^-)$
are also counted by $\perm(B^+)$.
%Its path indices are $I \cup \ol I$ and
In these families, the subfamilies of paths indexed by $I$ cover $Z_{I,I}$.
By [56],
these subfamilies correspond bijectively to
descent-free permutations of $Q_I$, with no elements circled.
On the other hand, the path families counted by $\perm(B^+)$
which are not counted by $\perm(B^-)$ are
those which contain paths having source $i$ and sink $\ol j$ for $j \geq 0$.
For each such path family $\pi$, define the path family $\sigma$ by doing the
following for each path $\pi_i$ with negative sink index $\ol j$.
\begin{enumerate}
\item Let $v$ be the latest vertex in the intersection of
$\pi_i$ and $\pi_{\ol i}$.
\item Swap the $v$-to-sink subpaths of $\pi_i$ and $\pi_{\ol i}$.
\end{enumerate}
If the path family $\pi$ has type $w \in \mfb k$,
then $\sigma$ has type $\varphi(w) \in \mfs k$.
(Explain why  all path families covering $Z_{I,I}$ appear here.)
By [56],
these subfamilies correspond bijectively to
descent-free permutations of $Q$, with circled elements corresponding
to paths which were modified to create $\sigma$.

By \cite[Prop.\,3.5]{SkanCCS},
the number of excedance-free $Q_I$-permutations equals the number of
descent-free $Q_I$-permutations.  The result now follows from

($\star$ Use the type $A$ interpretation of permanents in
terms of acyclic orientations.)

Each path family $\pi$ of type $e \in \mfb k$ covering $Z_{I\cup \ol I,I\cup \ol I}$
which is counted by $\det(B^-)$ must also be counted by
$\det(B^+)$.  Its path indices are $I \cup \ol I$, and
the subfamily of paths indexed by $I$ forms a chain in $Q$.
This chain contains no grounded element of $Q$ if $\pi$ contributes
to $\det(B^-)$, and at most one grounded element if $\pi$
contributes to $\det(B^+)$.
In the second case, if the chain contains a grounded element of $Q$,
then there exists exactly one other path family $\sigma$
of type $t \in \mfb k$ covering $Z_{I\cup \ol I,I\cup \ol I}$.
These two path families corresponding
to the same chain of $Q$ can be distinguished by circling the
grounded element of $Q$ in the case that the path family has type
$t \in \mfb k$.

By Theorem~\ref{t:perDBPB}, $\perm(B^-)$
counts the path families
\begin{equation}\label{eq:doublepathfam}
\pi = (\pi_{\ol k} \cdots \pi_{\ol 1}, \pi_1, \dotsc, \pi_k)
\end{equation}
%be a path family
covering $Z_{I \cup \ol I, I \cup \ol I}$ and containing no grounded path.
It is easy to see that the subfamily $(\pi_1,\dotsc, \pi_k)$
corresponding to (\ref{eq:doublepathfam}) covers the network
$Z_{I,I}$, contains no grounded paths,
and has sets of source and sink indices both equal to $[k]$.
Thus the tableau having these paths as entries has the claimed properties.
Conversely, each path family $(\rho_1,\dotsc,\rho_k)$ of type $v \in \mfs k$
covering $Z_{I,I}$ and posessing the claimed properties
%and containing no grounded paths
can be combined with its mirror image $(\rho_{\ol k}, \dotsc, \rho_{\ol 1})$,
where $\rho_{\ol j}$ is the unique path in $Z_{\ol I, \ol I}$
from source $\ol j$ to sink $v_{\ol j}$,
to create a path family
$(\rho_{\ol k}, \dotsc, \rho_{\ol 1}, \rho_1, \dotsc, \rho_k)$
covering $Z_{I \cup \ol I, I \cup \ol I}$ and containing no grounded path.
%where $\rho_{\ol j}$ is the mirror image of $\rho_j$,
%which covers $Z_{I \cup \ol I, I \cup \ol I}$.
%Clearly the subfamilies $(\pi_1,\dotsc,\pi_k)$
%correspond bijectively to $Z_{I,I}$-tableaux $U = (\pi_1,\dotsc,\pi_k)$
%which are row-closed, left row-strict, and containin no grounded paths.

On the other hand,
by Theorem~\ref{t:perDBPB}, $\perm(B^+)$
counts {\em all} path families (\ref{eq:doublepathfam})
%$\pi = (\pi_{\ol k} \cdots \pi_{\ol 1}, \pi_1, \dotsc, \pi_k)$
%be a path family
covering $Z_{I \cup \ol I, I \cup \ol I}$.
For each such path family of type $w \in \mfb k$, define
$v = |w_1| \cdots |w_k| \in \mfs k$
%by $v_i = |w_i|$ for all $i$,
and let $\rho = \rho(\pi) = (\rho_1, \dotsc, \rho_k)$
be the unique path family of type $v \in \mfs k$ covering $Z_{I,I}$.
Now create the $Z_{I,I}$-tableau
$U = (\rho_1,\dotsc,\rho_k)$
and for each negative component $w_i$ of $w$,
circle $\rho_i$.
By definition $U$ is left row-strict,
and it is clear that $\rho_i$ can be circled only if the sink index of $\pi_i$
is negative, i.e., only if $\pi_i$ is grounded.
Conversely, each tableau
$U = (\rho_1,\dotsc,\rho_k)$ of type $v \in \mfs k$
in which $\rho_{j_1}, \dotsc, \rho_{j_p}$
are circled corresponds to the unique path family (\ref{eq:doublepathfam})
in $Z_{I \cup \ol I, I \cup \ol I}$
of type $w = w_{\ol k} \cdots w_{\ol 1} w_1 \cdots w_k \in \mfb k$, where
\begin{equation*}
w_i = \begin{cases}
v_i &\text{if $i > 0$ and $\rho_i$ is not circled},\\
\ol{v_i} &\text{if $i > 0$ and $\rho_i$ is circled},
\end{cases}
\end{equation*}
and $w_{\ol k} \cdots w_{\ol 1} = \ol{w_k} \cdots \ol{w_1}$.

By (bijection result) such sequences correspond bijectively to
descent-free path tableaux

($\star$ Clean this up.)
Let $Q$ be the type-$\msfBC$ unit interval order.
$\perm(A^+)$ equals the number of
row-closed, left row-strict $\pi$-tableaux

\section{Old poset tableaux}

type-$\msfBC$ unit interval orders give an alternative to $F$-bitableaux.
Given a type-$\msfBC$ unit interval order $P$ on $n$ elements,
we fill a pair of (French) Young diagrams with these elements
%the positively indexed paths
%$\pi^+ \defeq (\pi_1,\dotsc,\pi_n)$
and call the resulting structure a {\em $P$-bitableau}.
%more specifically, a {\em $\pi$-bitableau}.  If $\pi$ has type $w \in \bn$,
%then we will also say that each $\pi$-bitableau has {\em type $w$}.
Define a {\em marked $P$-bitableau} $(U,V)$ to be a $P$-bitableau
in which all grounded elements appear in $U$,
and some subset of these is marked with asterisks.
For $(\lambda,\mu) \vdash n$
%a bipartition of $n$ with $\lambda \vdash a$ and
%$\mu \vdash n-a$, and $Z$ a zig-zag network of type $BC$,
let $\bitab(P,\lambda,\mu)$ be the set of all marked $P$-bitableaux $(U,V)$
of shape $(\lambda,\mu)$.
%Given a marked $\pi$-bitableau $(U,V)$ and $T \in \{ U,V \}$, let
%$T_i$ be the $i$th row of $T$, and let $T_{i,j}$ be the $j$th entry in row $i$.

%($\star$ Give examples of marked bitableaux of different shape and having
%some of the properties below, and a couple of entries $T_{i,j}$.)
($\star$ Give examples of marked $P$-bitableaux corresponding to earlier
used marked $F$-bitableaux?)

%($\star$ State that some sets of the form
%  (\epsilon\epsilon)^{\lambda,\mu}(\btc w1) =
%\# \{ (U,V) \in \bitab(Z,\lambda^\tr,\mu^\tr) \,|\,
%U, V \text{ both column-strict of type } e\}
%appearing in Theorem~\ref{t:epsiloneta}
%can be restated in terms of posets.)

\begin{cor}\label{c:epsilonetachipsi}
Let $w \in \bn$ be $\msfBC$-smooth with corresponding type-$\msfBC$
unit interval order $P$,
fix a bipartition $(\lambda,\mu)$ of $n$,
and define the sets $\mathcal S$, $\mathcal R$ of tableau properties
\begin{equation*}
\begin{gathered}
\mathcal S = \{ \text{row-semistrict}, \text{excedance-free} \},\\
\mathcal R = \{ \text{cyclically row-semistrict},
\text{record-free and row-semistrict} \}.
\end{gathered}
\end{equation*}
Then we have the following expressions for trace evaluations at $\btc w1$.
%induced one-dimensional characters in terms of
%the zig-zag network $Z = Z(w)$ as follows.
\begin{enumerate}[1.]
\item
$(\epsilon\epsilon)^{\lambda,\mu}(\btc w1) =
%    \begin{equation*}
\# \{ (U,V) \in \bitab(P,\lambda^\tr,\mu^\tr) \,|\,
U, V \text{ both column-strict }\}$.
%    \end{equation*}
\item  For any property $i \in \mathcal S$,
\begin{equation*}
(\epsilon\eta)^{\lambda,\mu}(\btc w1) =
\#\{(U,V) \in \bitab(P,\lambda^\tr,\mu) \,|\,
U \text{ column-strict}, V \text{ has property } i \}.
\end{equation*}
\item For any property $i \in \mathcal S$,
\begin{equation*}
(\eta\epsilon)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(P,\lambda,\mu^\tr) \,|\,
U \text{ has property } i, 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(P,\lambda,\mu) \,|\,
U \text{ has property } i, V \text{ has property } j \}.
\end{equation*}
\item
$(\chi\chi)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(P,\lambda,\mu) \,|\,
U, V \text{ both standard } \}$.
\item For any properties $i, j \in \mathcal R$,
\begin{equation*}
(\psi\psi)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(P,\lambda,\mu) \,|\,
U \text{ has property } i, V \text{ has property } j \}.
\end{equation*}
\end{enumerate}
\end{cor}

\section{Old $\psi$ tableaux interpretations}

\begin{thm}\label{t:psi}
Let $w \in \bn$ be $\msfBC$-smooth with zig-zag network $F_w$ and
%  Let $w \in \bn$ be $\msfBC$-smooth,
fix a 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(Z,\lambda,\mu) \,|\,
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}

\section{Old $\epsilon,\eta$ tableaux interpretations}

\begin{thm}\label{t:epsiloneta}
Let $w \in \bn$ be $\msfBC$-smooth,
%(what patterns?),
%let $B$ be the path matrix of
%let $Z = Z(w)$,
%let $P = P(w)$,
fix a bipartition $(\lambda,\mu)$ of $n$,
and define the set $\mathcal S$ of three 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*}
\mathcal S = \{ \text{ row-semistrict of type $e$},
\text{excedance-free of type $e$},
\text{row-closed and left-row-strict } \}.
\end{equation*}
Then we may express the evaluations at $\btc w1$ of
induced one-dimensional characters in terms of
the zig-zag network $Z = Z(w)$ as follows.
\begin{enumerate}[1.]
\item
$(\epsilon\epsilon)^{\lambda,\mu}(\btc w1) =
%    \begin{equation*}
\# \{ (U,V) \in \bitab(Z,\lambda^\tr,\mu^\tr) \,|\,
U, V \text{ both column-strict of type } e\}$.
%    \end{equation*}
\item  For any property $i \in \mathcal S$,
\begin{equation*}
(\epsilon\eta)^{\lambda,\mu}(\btc w1) =
\#\{(U,V) \in \bitab(Z,\lambda^\tr,\mu) \,|\,
U \text{ column-strict of type } e, V \text{ has property } i \}.
\end{equation*}
\item For any property $i \in \mathcal S$,
\begin{equation*}
(\eta\epsilon)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(Z,\lambda,\mu^\tr) \,|\,
U \text{ has property } i, V \text{ column-strict of type }e \}.
\end{equation*}
\item For any properties $i, j \in \mathcal S$,
\begin{equation*}
(\eta\eta)^{\lambda,\mu}(\btc w1) =
\# \{ (U,V) \in \bitab(Z,\lambda,\mu) \,|\,
U \text{ has property } i, V \text{ has property } j \}.
\end{equation*}
\end{enumerate}
\end{thm}

\section{Old rational function proof}

Let $\pi = (\pi_1,\dotsc,\pi_n)$ be the path family
such that
the $m$ paths covering the above edges are $\pi_{

Observe that for $t=1$, the star network $F_{[c_1,d_1]}$
graphically represents the Kazhdan--Lusztig basis element
\begin{equation*}
\wtc{s_{[c_1,d_1]}}q = \sum_{v \leq s_{[c_1,d_1]}} \nTksp T_v.
\end{equation*}
Now assume that for $j = 1,\dotsc,t$,
we have an analog of the formula (\ref{eq:ratlcw})
for concatenations of $j$ star networks,
i.e., where the product in the denominator includes only
intervals in $D_j \defeq \{ [c_i,d_i] \,|\, i \leq j \}$.
%        define $D_j$ to be the set of intervals
%        and
Consider the concatenations
\begin{equation*}
G_{t-1} =  F_{[c_1,d_1]} \bullet \cdots \bullet F_{[c_{t-1},d_{t-1}]},
\qquad\qquad
G_{t-1} \circ F_{[c_t,d_t]},
\end{equation*}
%Then $G_{t-1}$ and $G_{t-1} \circ F_{[c_t,d_t]}$
which graphically represent the
$H_{[h,l]}(q)$ elements
\begin{equation}\label{eq:ratls}
\frac{\wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_{t-1},d_{t-1}]}}q}
\prod_{\substack{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\\text{ in } D_{t-1}}}}
\nTksp \nTksp |[c_i,d_i] \cap [c_j,d_j]|_q!},
\qquad \qquad
\frac{\wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_{t},d_{t}]}}q}
\prod_{\substack{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\\text{ in } D_{t-1}}}}
\nTksp \nTksp |[c_i,d_i] \cap [c_j,d_j]|_q!},
\end{equation}
%       where the product in both fractions include
%       only intervals in $D_{t-1}$.
%       and $G_{t-1} \circ F_{[c_t,d_t]}$ graphically represents the
%       product of (\ref{eq:ratlt-1}) and $\wtc{[c_t,d_t]}q$.
%indexed by $1, \dotsc, t-1$.
%in $P_{t-1}$
%the poset
%induced by intervals $[c_1,d_1],\dotsc,[c_{t-1},d_{t-1}]$.
%Suppose first that
If $G_{t-1} \circ F_{[c_t,d_t]}$
is a simple directed graph, i.e.,
for all $[c_i,d_i] \precdot [c_t,d_t]$
we have $|[c_i,d_i] \cap [c_t,d_t]| \leq 1$,
then $G_{t-1} \circ F_{[c_t,d_t]} = G_{t-1} \bullet F_{[c_t,d_t]}$
and the second fraction in (\ref{eq:ratls}) equals (\ref{eq:ratlcw}).
If $G_{t-1} \circ F_{[c_t,d_t]}$
is not a simple directed graph, then for some index set $I$,
vertices $\{ x_i \,|\, i \in I \}$ in $G_{t-1}$ corresponding to
$\{[c_i,d_i] \precdot [c_t,d_t] \,|\, i \in I\}$
are adjacent to the vertex $y$ corresponding to interval $[c_t,d_t]$,
with $m(x_i,y)$ edges from $x_i$ to $y$.
%of $F_{[c_t,d_t]}$
%a central vertex $x_i$ from which $m(x_i)$ edges terminate

These networks graphically represent the
$H_{[h,l]}(q)$ elements
\begin{equation*}
\qquad
\frac{\wtc{s_{[c_1,d_1]}}q \cdots \wtc{s_{[c_{t-1},d_{t-1}]}}q}
{\displaystyle{\prod_{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]}}
\nTksp \nTksp |[c_i,d_i] \cap [c_j,d_j]|_q!}
\wtc{s_{[c_t,d_t]}}q.
\end{equation*}

%          .
%          F_{[c_1,d_1]} \bullet \cdots \bullet F_{[c_t,d_t]}.
(\ref{eq:ratlcw}) as it is stated.
Let $G = F_{[c_1,d_1]} \bullet \cdots \bullet F_{[c_{r-1},d_{r-1}]}$
and consider the graph $G \bullet F_{[c_r,c_r]}$.

It follows that for any sequence
$([c_1,d_1], \dotsc, [c_t,d_t])$
of subintervals of $[h,l]$, the star network
$F_{[c_1,d_1]} \circ \cdots \circ F_{[c_t,d_t]}$ graphically represents
the numerator of (\ref{eq:ratlcw}).  Now consider
the standard concatenation symbols $\circ$
from right to left,
and let $G_{i+1}$ be the
(modified) concatenation
$F_{[c_{i+1},d_{i+1}]} \bullet \cdots \bullet F_{[c_t,d_t]})$.
If $F_{[c_i,d_i]} \circ G_{i+1}$ is a simple directed graph,
then $F_{[c_i,d_i]} \circ G_{i+1} = F_{[c_i,d_i]} \bullet G_{i+1}$
is a multigraph in which the central vertex $x$ of $F_{[c_i,d_i]}$
and some vertex $y$ in $G_{i+1}$
$\star$ Find result in Clearwater-Skandera~[15] ?

\section{Pattern avoidance Hell}

the short one-line notation of $w$ avoids the signed patterns
(\ref{eq:26signedpatterns}) and $\ol2\ol1$.
From this list we may omit the ten signed patterns
\begin{equation}\label{eq:bavoidp}
1\ol3\ol2, \ol2\ol1\ol3, 2\ol3\ol1, \ol31\ol2,
\ol3\ol2\ol1, \ol3\ol21, \ol32\ol1, 3\ol2\ol1,
\ol3\ol4\ol1\ol2, \ol34\ol12,
%2\ol431, \ol3412, \ol4231,
\end{equation}
which themselves have subwords matching the signed pattern $\ol2\ol1$,
to obtain an equivalent list of
%this leaves us with a total of
seventeen signed patterns.
%In type $\mathsf C$,
The type-$\mathsf C$ Schubert variety
%$\schub w$
$\Cschub w$
is smooth if and only if
the short one-line notation of $w$ avoids the signed patterns
(\ref{eq:26signedpatterns}) and $1\ol2$.
%Since the nine signed patterns
From this list we may omit the ten signed patterns
\begin{equation}\label{eq:cavoidp}
\ol12\ol3, 1\ol2\ol3, 12\ol3, 1\ol3\ol2,
\ol21\ol3,
2\ol1\ol3, 2\ol3\ol1,
\ol31\ol2, 2\ol431,
\end{equation}
which themselves have subwords matching the signed pattern $1\ol2$,
to obtain an equivalent list of
%themselves have subwords which match the signed pattern $1\ol2$,
%this again leaves us with a total of
seventeen signed patterns.
%($\star$ List the two patterns and cite the book.)
%These conditions are stated in terms of the avoidance
%of dozens of patterns and signed patterns listed
%($\star$ List them.)
%($\star$ Do we need more details, or is this sufficient?)
%\item $X(w)$ is rationally smooth in types $\mathsf B$ and $\mathsf C$
%  if the short one-line notation of $w$
%avoids the 26 signed patterns on
%or if the long one-line notation of $w$
%must avoid the 25 unsigned patterns on \cite[p.\,210]{BilleyLak}.
%\item $X(w)$ is smooth in
%  in type $\mathsf B$
%  if the short one-line notation of $w$
%must avoid the 26 signed patterns on \cite[p.\,210]{BilleyLak}
%and $\ol2 \ol1$,
%or equivalently, the long one-line notation of $w$
%must avoid the 25 unsigned patterns on \cite[p.\,210]{BilleyLak}
%and $3412$.
%\item In type $\mathsf C$,
%the short one-line notation of $w$
%must avoid the 26 signed patterns on \cite[p.\,210]{BilleyLak}
%and $1 \ol2$,
%or equivalently, the long one-line notation of $w$
%must avoid the 25 unsigned patterns on \cite[p.\,210]{BilleyLak}
%and $4231$.

We will be interested in those elements $w \in \bn$ for which
the type-$\mathsf B$ and type-$\mathsf C$ Schubert varieties $X(w)$
are {\em simultaneously} smooth.
%The above arguments characterize
%such elements by the avoidance of eleven signed patterns;
This condition is equivalent to the avoidance of just two unsigned patterns.
%On the other hand, there is a much shorter description of elements $w$
%for which the Schubert variety $X(w)$ is smooth in {\em both} types
%$\mathsf B$ and $\mathsf C$.
\begin{prop}\label{p:bcavoid}
For $w \in \bn$,
the type-$\mathsf B$ and type-$\mathsf C$ Schubert varieties
$X(w)$ are simultaneously smooth
%in types $\mathsf B$ and $\mathsf C$ simultaneously
if and only if the long one-line notation of $w$
avoids the two unsigned patterns $3412$, $4231$.
\end{prop}
\begin{proof}
%  By ($\star$ \cite[Thms.\,4.2, 6.1]{BilleyPattern}
%  or better: just cite the book,
%  where the proof appeared for the first time, anyway.)
The type-$\mathsf B$ and type-$\mathsf C$ Schubert varieties
%$X(w)$ is smooth in types
%$\mathsf B$ and $\mathsf C$
are simultaneously smooth if and only if
%  ($\star$ Flesh this out.)
the short one-line notation of $w$ avoids the twenty-six signed patterns
%(26 signed patterns)
and $\ol2\ol1$ and $1 \ol2$.
Adding these last two signed patterns to the list and removing those
appearing in (\ref{eq:bavoidp}), (\ref{eq:cavoidp}),
%signed patterns already containing these,
we obtain
%This boils down to the
the eleven signed patterns
\begin{equation}\label{eq:avoidlist}
\ol2\ol1, 1\ol2, 3\ol21, \ol2\ol431, \ol3412, 34\ol12, 3412, 4\ol13\ol2,
\ol4231, 423\ol1, 4231.
\end{equation}

Now suppose that the long one-line notation of $w$ contains a subword
$v = v_1v_2v_3v_4$ which matches the pattern $3412$.
Let the absolute values of these letters be $a < b < c < d$,
and let $k$ be the number of negative letters in $v$.
Consider the signed pattern consequences of $w$ containing $v$
for  $k = 0,\dotsc,4$.
%, then $v$ is one of the words
%  cdab, cd\ol ab, cd\ol ba, bd\ol ca, bc\ol da, cd\ol b\ola
\begin{enumerate}
\item[($k=0$)] We have
%If $0$ letters of $v$ are negative, then
$v = cdab$.
If $v$ appears in positive positions then
it matches the signed pattern $3412$.
%$w_1 \cdots w_n$
%contains the signed pattern $3412$.
If $v_1 = c$ is the only letter of $v$ to appear in a negative position,
then $w_1 \cdots w_n$ contains
%$\ol 3412$, $4\ol312$, $41\ol32$, or $412\ol3$,
$\ol cdab$, $d\ol cab$, $da\ol cb$, or $dab\ol c$,
which contains a subword matching
%matches the signed pattern $\ol3412$ or contains a
%subwords matching
the signed pattern $\ol3412$, $3\ol21$, $1\ol2$, $1\ol2$, respectively.
If both $c$ and $d$ appear in negative positions, then $\ol d$, $\ol c$
appear in positive positions, matching the signed pattern $\ol2\ol1$.
\item[($k=1$)]
%\noindent (Case $1$) Assume that
%  If $v$ contains exactly one negative letter,
%$v_3 < 0$.
%then $v$ is one of the words
%we have
%$v = $
%the letters are
%$34\ol12$, $34\ol21$, $24\ol31$, $23\ol41$, or $23\ol11$.
We have $v \in \{cd\ol ab, cd\ol ba, bd\ol ca, bc\ol da\}$.
%or $23\ol11$.
%In the last case, $23$ must lie on the left side of $w$ and $\ol3\ol2$
%lies on the right, matching the pattern $\ol2\ol1$.
Suppose $v$ lies on the right side of $w$.
If $v = cd\ol ab$ then it matches the signed pattern $34\ol12$;
if $v$ is one of the other three words
%($\star$ what?),
% \in \{ cd\ol ba, bd\ol ca, d \ol a b \olc\}$,
then on the right side of $w$ we have
$v_2v_3v_4 \in \{d \ol ba, d \ol ca \}$
matching the pattern $3\ol21$,
or $v_2v_3 = c\ol d$ matching the pattern $1 \ol  2$.
%or $v_3v_4 = b\ol c$ matching the pattern $1 \ol  2$.
Suppose that only $v_1$ lies on the left side of $w$.
If $v = cd\ol ab$, then one of the words
$\ol c d\ol ab$,
$d \ol c \ol a b$,
$d \ol a \ol c b$,
$d \ol a b \ol c$
lies on the right side of $w$.
These contain subwords $\ol c \ol a$, $d \ol c b$, $b \ol c$
matching the signed patterns
$\ol2\ol1$, $3\ol21$, $1\ol2$, respectively.
If $v$ is one of the other three words
%($\star$ what?),
% \in \{ cd\ol ba, bd\ol ca, d \ol a b \olc\}$,
then on the right side of $w$ we have
$v_2v_3v_4 \in \{d \ol ba, d \ol ca \}$
matching the pattern $3\ol21$,
or $v_2v_3 = c\ol d$ matching the pattern $1 \ol  2$.
%or $v_3v_4 = b\ol c$ matching the pattern $1 \ol  2$.
If both $v_1$, $v_2$ appear on the left side of $w$,
then $\ol{v_2v_1}$
appears on the right and matches the signed pattern $\ol2 \ol1$.
($\star$ Shorten this by starting with $2$ letters on the left.)
%If $v = d \ol a b \ol c$, then $b \ol c$ lies on the right
\item[$(k=2)$]
%If exactly $0$, $1$, or $2$ letters of $v$ appear on
%the left side of $w$, then
If at least the two negative letters $v_3v_4$ appear
on the right of $w$, then these match the signed pattern $\ol2 \ol1$;
if at least the two positive letters $v_1v_2$ appeear
on the left of $w$, then $\ol{v_2} \ol{v_1}$ appear on the right
and also match the signed pattern $\ol2 \ol1$.
\item[$(k\geq3)$] We have that $\ol{v_4v_3v_2v_1}$ is also a subword
of $w$ which matches the pattern $3412$.
It has $0$ or $1$ negative letters and is included in the
cases $k=0,1$.
%then we have subwords of $w$ which match the patterns ?,?,?,?
%If $v_1v_2$ lies on the left side of $w$, then (again) we have $\ol4\ol3$
%on the right, which matches the pattern $\ol2\ol1$.
%Now suppose that $v_3$, $v_4$ are the only negative letters...
%Now suppose that $v_4$ is the only positive letter...
%Now suppose that all four letters are negative....
\end{enumerate}

($\star$ Include the possibility of repeated letters.)

Suppose that the long one-line notation of $w$ contains a subword
$v = v_1v_2v_3v_4$ which matches the pattern $4231$.
Again, let the absolute values of these letters be $a < b < c < d$,
and let $k$ be the number of negative letters in $v$.
Consider the signed pattern consequences of $w$ containing $v$
for  $k = 0,\dotsc,4$.
\begin{enumerate}
\item[($k=0$)] We have
$v = dbca$.
If $v$ appears in positive positions then
it matches the signed pattern $4231$.
If $d$ is the only letter of $v$ to appear in a negative position,
then $w_1 \cdots w_n$ contains
%$\ol 3412$, $4\ol312$, $41\ol32$, or $412\ol3$,
$\ol dbca$, $b\ol dca$, $bc\ol da$, or $bca\ol d$,
which matches the signed pattern
%have subwords matching
%the signed patterns
or contains a subword matching the pattern $1 \ol2$.
%$1\ol2$, $1\ol2$, $1\ol2$, respectively.
If only $d$ and $b$ appear in negative positions, then $\ol b$, $\ol d$
appear in positive positions.  If these appear before $c$, then the subword
$\ol b \ol d c a$ matches the pattern $\ol2 \ol4 3 1$.
Otherwise, $\ol d$ appears after $c$ and the subword $c \ol d$ matches
the signed pattern $1\ol2$.
If $d$, $b$, and $c$ appear in negative positions, then $\ol c$, $\ol b$
appear in positive positions, matching the signed pattern $\ol2 \ol1$.
\item[($k=1$)]
We have $v \in \{dbc\ol a, dac \ol b, dab \ol c, cab\ol d\}$.
If $v$ lies on the right side of $w$,
then it matches the signed pattern $423\ol1$ or contains a subword
matching the signed pattern $1\ol2$.
Suppose that only $v_1$ lies on the left side of $w$.
If $v = dbc\ol a$, then the right side of $w$ contains $\ol d \ol a$ which
matches the pattern $\ol2 \ol1$, or it contains $c \ol d$ which
matches the pattern $1\ol2$.  If $v$ is one of the other three words, then
$v_2v_4$ matches the pattern $1\ol2$.
Now suppose that only $v_1$, $v_2$ lie on the left side of $w$.
If $v = dbc\ol a$, then again, the right side of $w$ contains
$\ol d \ol a$ which matches the pattern $\ol2 \ol1$,
or it contains $c \ol d$ which matches the pattern $1\ol2$.
If $v = dac\ol b$, then the right side of $w$ contains
$\ol d \ol b$ which matches the pattern $\ol2 \ol1$,
or it contains $c \ol d$ which matches the pattern $1\ol2$.
If $v$ is one of the other two words, then
$v_3v_4 \in \{ b\ol c, b\ol d\}$ matches the pattern $1 \ol2$.
If $v_1$, $v_2$, $v_3$ all appear on the left then $\ol{v_3}\ol{v_2}$ appears
on the right and matches the pattern $\ol2 \ol1$.
\item[($k=2$)]
We have $v \in \{ d \ol a c \ol b, c \ol a b \ol c, c \ol a b \ol d,
c \ol b a \ol d, d \ol b a \ol c, b \ol c a \ol d \}$.
If $v$ is anything except $d \ol a c \ol b$ and if $v_3v_4$ appear on the
right of $w$, then these match the signed pattern $1\ol2$.

\end{enumerate}

($\star$ Repeat arguments for this pattern.)
%these eleven signed patterns (\ref{eq:avoidlist}), including $3412$ and $4231$,
%then certainly the long one-line notation of $w$ avoids the ordinary patterns
%$3412$ and $4231$.

%Now observe that for any signed pattern
%$c = c_1 \cdots c_k$
%if a subword $b$ of the short one-line notation of $w$ matches $c$,
%then a subword $b'$ of the long one-line notation of $w$
%matches the pattern $\ol{c_k} \cdots \ol{c_1} c_1 \cdots c_k$.

%in (\ref{eq:avoidist})
Now we claim that
for each signed pattern $c = c_1 \cdots c_k$,
in (\ref{eq:avoidlist}),
if a subword
of the short one-line notation of $w$ matches $c$,
then a subword of the long one-line notation of $w$ matches $3412$ or $4231$.
To see this, observe that if a subword of $w_1 \cdots w_n$
matches the signed pattern $c$,
then a subword of $w_{\ol n} \cdots w_{\ol 1} w_1 \cdots w_n$
matches the unsigned pattern
$c' = \ol{c_k} \cdots \ol{c_1} c_1 \cdots c_k$.
It is easy to see that for each signed pattern $c$
in (\ref{eq:avoidlist}), an unsigned subpattern $b$ of $c'$ is equivalent to
%then a subword $w'$ of the long one-line notation of $w$ matches
%the ordinary pattern
%We claim that for each signed pattern $c$
%in (\ref{eq:avoidlist}) and the corresponding ordinary pattern $c''$,
%a subword $w''$ of $w$ matches a subword $b$ of $c''$,
%of this
%$b' = \ol{b_k} \cdots \ol{b_1} b_1 \cdots b_k$
%= w_{\ol{i_k}} \cdots w_{\ol{i_1}} w_{i_1} \cdots w_{i_k}$
%and finally, matching $w''$ is equivalent to matching
an unsigned pattern $d \in \{3412, 4231\}$.
Excluding the signed patterns $c = 3412, 4231$ themselves,
these patterns $c, b, d$ are
%The table
\hhhsp
\begin{center}
\newcolumntype{R}{>{$}c<{$}}
\begin{tabularx}{113.2mm}{|R|R|R|R|R|R|R|R|R|R|}%
\hline
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& \ol2 \ol1
& 1 \ol2
& 3 \ol2 1
& \ol2\ol431
& \ol3412
& 34\ol12
& 4\ol13\ol2
& \ol4231
& 423\ol1\\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
\hline
%& & & & & & & & & \\
%& & & & & & & & & \\
%& & & & & & & & & \\
%& \ntnsp 12\ol2 \ol1 \ntksp
%& 2\ol11 \ol2
%& \ol12\ol33 \ol2 1
%& \ol1\ol342 \ol2\ol431
%& \ol2\ol1\ol43\ol3412
%& \ol21\ol4\ol3 34\ol12
%& 2\ol31\ol4 4\ol13\ol2
%& \ol1\ol3\ol24 \ol4231
%& 1\ol3\ol2\ol4 423\ol1\\
%& & & & & & & & & \\
%& & & & & & & & & \\
%& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& 12 \ol2 \ol1
& 2\ol1 1 \ol2
& 23 \ol2 1
& 4231
& 3412
& 34\ol12
& 4\ol13\ol2
& 4231
& 423\ol1\\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
\hline
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& 3412
& 4231
& 3412
& 4231
& 3412
& 3412
& 4231
& 4231
& 4231\\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
\hline
\end{tabularx}\; .
\end{center}
\ssp

\end{proof}

\section{tabular}

& & & & & & & & &\\
& & & & & & & & &\\
& & & & & & & & &\\
& & & & & & & & &\\
& & & & & & & & &\\
& & & & & & & & &\\
\hline
& & & & & & & & &\\
& & & & & & & & &\\
& & & & & & & & &\\
& & & & & & & &\\
& & & & & & & &\\
& & & & & & & &\\
\hline
& & & & & & & &\\
& & & & & & & &\\
& & & & & & & &\\
& & & & & & & &\\
& & & & & & & &\\
& & & & & & & &\\
\hline

\section{pattern avoidance and type bc uios}

These correspond bijectively
to elements of $\bn$ avoiding ($\star$ what pattern?) as follows.
\begin{prop}
Let $w \in \bn$ avoid the patterns ($\star$ which?) and define
the type-$\msfBC$ unit interval order $P(w)$ as recorded in a single
($\star$ State the bijection between $\msfBC$-$312$-avoiding permutations
and type-$\msfBC$ unit interval orders, using Hessenberg functions.)
\end{prop}

\section{Every function type B q=1}

\begin{prop}\label{p:everysymmfnbc}
%  Let $Q$ be an integral domain and let $F$ be its field of fractions.
Every symmetric function in $\Lambda_n(x,y)$
%  $\mathbb Z[q] \otimes \Lambda_n(x,y)$
has the form $Y(D)$ for some element $g \in \mathbb Q[\bn]$.
%  $g \in \mathbb Q(q)[\bn]$.
\end{prop}

\begin{proof}
%    Now consider the second statment.
Fix a symmetric function in $\Lambda_n(x,y)$,
and expand it in the plethystic power sum basis
as $\sum_{(\lambda,\mu) \vdash n} a_{\lambda,\mu} p_\lambda^+(x,y)p_\mu^-(x,y)$.
Then choose one representative $w_{\lambda,\mu}$ of each conjugacy class of $\bn$
and define the $\qbn$
%group algebra
\begin{equation*}
g = \sum_{(\alpha,\beta) \vdash n}
\frac{a_{\alpha,\beta}}{z_\alpha z_\beta 2^{\ell(\alpha) +\ell(\beta)}}
\end{equation*}
By (\ref{eq:iotadef}) and (\ref{eq:omegabc}) we have
\begin{equation*}
\begin{aligned}
Y(D) &= \sum_{(\lambda,\mu) \vdash n}
\iota^{\lambda,\mu}
\Big( \sum_{(\alpha,\beta) \vdash n}
\frac{a_{\alpha,\beta}}{z_\alpha z_\beta 2^{\ell(\alpha) +\ell(\beta)}}
(w_{\alpha,\beta})\Big) p_\lambda^+(x,y)p_\mu^-(x,y)\\
&= \sum_{(\lambda,\mu) \vdash n}
\frac{a_{\lambda,\mu}\iota^{\lambda,\mu}(w_{\lambda,\mu})}
{z_\lambda z_\mu 2^{\ell(\lambda) +\ell(\mu)}}
&= \sum_{(\lambda,\mu) \vdash n} a_{\lambda,\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}

\section{Old Hecke}

%($\star$ Decide what to do with this section.
%Define Hecke algebra and Kazhdan--Lusztig basis
%generically in terms of a Coxeter group $W$?
%State the type-$\msfA$ and type-$\msfBC$ presentations in full detail
%only if it turns out that we need them?)
%$\star$ Can we leave out most of this section and just define
%the Kazhdan--Lusztig basis elements for $w$ \avoidingp?
%Do we need this section at all?

%Following [36], we
Define the type-$\msfA$ Hecke algebra $\hnq$ of $\sn$
%$H_{[\ol n,n]}(q)$ of $\snn$
to be the noncommutative $\zqq$-algebra with unit $T_e$,
generated by $T_{s_1}, \dotsc, T_{s_{n-1}}$,
%$T_{s_{\ol{n-1}}}, \cdots, T_{s_{\ol 1}}, T_{s_0}, T_{s_1}, \dotsc, T_{s_{n-1}}$,
subject to relations
\begin{equation*}
\begin{alignedat}2
%    T_t^2 &= (q'-1)T_t + q', &\qquad & \\
T_{s_i}^2 &= (q-1)T_{s_i} + q &\qquad &\text{for $i = 1,\dotsc,n-1$}, \\
%    T_tT_{s_1}T_tT_{s_1} &= T_{s_1}T_tT_{s_1}T_t &\qquad &\\
%    T_tT_{s_i} &= T_{s_i}T_t &\quad &\text{for $i=2,\dotsc,n-1$,}\\
T_{s_i}T_{s_j} &= T_{s_j}T_{s_i} &\qquad &\text{for $|i-j| \geq 2$}, \\
T_{s_i}T_{s_j}T_{s_i} &= T_{s_j}T_{s_i}T_{s_j} &\qquad &\text{for $|i-j| =1$}.
\end{alignedat}
\end{equation*}
For $w \in \sn$ and
$w = s_{i_1} \cdots s_{i_\ell}$ a reduced expression, we define
$T_w = T_{s_{i_1}} \cdots T_{s_{i_\ell}}$.  The set $\{ T_w \,|\, w \in \sn \}$
forms a natural basis for $\hnq$ as a $\zqq$-module.

%\ssec{The type-$\msfBC$ Hecke algebra}
%While it might seem natural to define the Hecke algebra
%of $\bn$
%to be the subalgebra of $H_{2n}(q)$
%generated by
%  T_t = T_{s'_0} \defeq T_{s_0}, \qquad
%T_{s'_i} \defeq T_{s_i}T_{s_{\ol i}} \mbox{ for } i = 1, \dotsc, n-1,
%this definition would lead to a relation
%  T_{s'_i}^2 = T_{s_i}^2T_{s_{\ol i}}^2
%  = (q-1)^2T_{s'_i} + q(q-1)(T_{s_i} + T_{s'_i}) + q^2 T_e,
%which has a right-hand-side not belonging to the subalgebra
%generated by the elements (\ref{eq:temphgenerators}).

Similarly, define $\hbnq$
to be the algebra generated over $\zqq$
by the symbols $T_t = T_{s'_0} = T_{s_0}, T_{s'_1}, \dotsc, T_{s'_{n-1}}$
%%in (\ref{eq:temhgenerators})
and the relations
%These generators satisfy the relations
\begin{equation*}
\begin{alignedat}2
%    T_t^2 &= (q'-1)T_t + q', &\qquad & \\
T_{s'_i}^2 &= (q-1)T_{s'_i} + q &\qquad &\text{for $i = 0,\dotsc,n-1$}, \\
T_tT_{s'_1}T_tT_{s'_1} &= T_{s'_1}T_tT_{s'_1}T_t &\qquad &\\
%    T_tT_{s_i} &= T_{s_i}T_t &\quad &\text{for $i=2,\dotsc,n-1$,}\\
T_{s'_i}T_{s'_j} &= T_{s'_j}T_{s'_i} &\qquad &\text{for $i,j \geq 0$ and $|i-j| \geq 2$}, \\
T_{s'_i}T_{s'_j}T_{s'_i} &= T_{s'_j}T_{s'_i}T_{s'_j} &\qquad &\text{for $i,j \geq 1$ and $|i-j| =1$}.
\end{alignedat}
\end{equation*}
%If $s'_{i_1} \cdots s'_{i_\ell}$ is a reduced expression for $w \in \bn$,
%we define $T_w = T_{s'_{i_1}} \cdots T_{s'_{i_\ell}}$.