# The Chromatic Quasisymmetric Class Function of a Digraph

Jacob A. White  School of Mathematical and Statistical Sciences
University of Texas - Rio Grande Valley
Edinburg, TX 78539

###### Abstract.

We introduce a quasisymmetric class function associated to a group acting on a
double poset or on a directed graph. The latter is a generalization of the
chromatic quasisymmetric function of a digraph introduced by Ellzey, while the
latter is a generalization of a quasisymmetric function introduced by
Grinberg. We prove representation-theoretic analogues of classical and recent
results, including $F$-positivity, and combinatorial reciprocity theorems. We
also deduce results for orbital quasisymmetric functions. We also study a
generalization of the notion of strongly flawless sequences.

###### Key words and phrases:

Chromatic Polynomials, Quasisymmetric Functions, Poset partitions, Group
actions

###### 1991 Mathematics Subject Classification:

05E05, 05E18

##  1\. Introduction

Given a graph $G$, let $\mathfrak{G}$ be a subgroup of the automorphism group
of $G$. Then $\mathfrak{G}$ acts on the set of $k$-colorings of $G$. If we
$\chi_{\mathfrak{G}}(G,k)$ denote the number of orbits of this action, then
the resulting function is a polynomial in $k$, called the _orbital chromatic
polynomial_ and studied by Cameron and Kayibi [5]. Jochemko [13] found a
combinatorial reciprocity theorem by giving a combinatorial interpretation to
$(-1)^{n}\chi_{\mathfrak{G}}(G,-k)$.

Similarly, given a poset $P$, a subgroup $\mathfrak{G}$ of the automorphism
group of $P$ acts on the set of order-preserving maps
$\varphi:P\to\\{1,\ldots,k\\}$. If we let $\Omega_{\mathfrak{G}}(P,k)$ denote
the number of orbits of this action, we obtain the orbital order polynomial
that was introduced by Jochemko [13], who proved a combinatorial reciprocity
theorem for $\Omega_{\mathfrak{G}}(P,k)$. These results were later generalized
to quasisymmetric functions associated to a double poset by Grinberg [10]. One
of our primary interests is proving combinatorial reciprocity theorems for
orbital polynomial invariants associated to combinatorial objects.

Stapledon [19] studied the equivariant Ehrhart quasipolynomial of a polytope.
Let $\mathfrak{G}$ be a finite group acting linearly on a lattice $M^{\prime}$
of rank $n$, and let $P$ be a $d$-dimensional $\mathfrak{G}$-invariant lattice
polytope. Let $M$ be a translation of the intersection of the affine span of
$P$ and $M^{\prime}$ to the origin, and consider the induced representation
$\rho:\mathfrak{G}\to GL(M)$ If $\chi(m)$ is the permutation character
associated to the action of $G$ on the lattice of points in the $m$th dilate
of $P$, then $\chi(m)$ is a quasipolynomial in $m$ whose coefficients are
elements of $R(\mathfrak{G})$, the ring of virtual characters of
$\mathfrak{G}$. Stapledon proved several results concerning the equivariant
Ehrhart quasipolynomial.

Motivated by these past results, we study quasisymmetric class functions.
These are class functions associated to the symmetry group $\mathfrak{G}$ of a
combinatorial object, whose values are quasisymmetric functions. Equivalently,
they are quasisymmetric functions whose coefficients are class functions. If
we let $\mathfrak{G}$ be the trivial group, then we obtain ordinary
quasisymmetric functions (and should re-derive classical results). In fact, we
can always obtain the ordinary quasisymmetric function by evaluating all
characters at the identity element. We can also obtain corresponding orbital
quasisymmetric functions, and various polynomial specializations.

The goal of this paper is to study a quasisymmetric class function
generalization of the $D$-partition enumerator of a double poset and of the
chromatic polynomial of a directed graph. The former is an class function
generalization of an invariant introduced by Grinberg, which in turn is a
generalization of the labeled $P$-partition enumerator studied by Gessel [9].
We define double posets and related terminology in Section 3. Given a double
poset $D$ on a set $N$, let $\mathfrak{G}$ be a subset of the automorphism
group of $D$. Then $\mathfrak{G}$ acts on the set of $D$-partitions. For
$\mathfrak{g}\in\mathfrak{G}$, we define

$\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum\limits_{\sigma:\mathfrak{g}\sigma=\sigma}\prod_{v\in
N}x_{\sigma(v)}$

where we are summing over $D$-partitions fixed by $\mathfrak{g}$. Then
$\Omega(D,\mathfrak{G},\mathbf{x})$ is a $\operatorname{QSYM}$-valued class
function.

Stanley introduced the chromatic symmetric function [18], a symmetric function
generalization of the chromatic polynomial. This has been generalized to a
chromatic quasisymmetric function by Shareshian and Wachs [16] and to directed
graphs by Ellzey [7]. We will study a class function generalization of
Ellzey’s invariant, defined more explicitly in Section 4. Much like the
generalization of Shareshian and Wachs, our invariant has an extra variable
$t$: our invariant is a class function that takes on values in the ring of
quasisymmetric functions over the field $\mathbb{C}(t)$. Given a digraph $G$
on a set $N$, let $\mathfrak{G}$ be a subset of the automorphism group of $G$.
Then $\mathfrak{G}$ acts on the set of proper colorings of the underlying
undirected graph. Ellzey defines a statistic $\operatorname{asc}(f)$ for a
coloring. We show that this statistic is also $\mathfrak{G}$-invariant. For
$\mathfrak{g}\in\mathfrak{G}$, we define

$\chi(G,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum\limits_{f:\mathfrak{g}f=f}t^{\operatorname{asc}(f)}\prod_{v\in
N}x_{\sigma(v)}$

where we sum over proper colorings of $G$. Much like in the case of double
posets, the resulting invariant is a class function whose values are
quasisymmetric functions over $\mathbb{C}[t]$.

Our primary interest is to study generalizations of $F$-positivity results,
inequalities, and combinatorial reciprocity theorems. Let
$C(\mathfrak{G},\operatorname{QSYM})$ be the set of class functions with
values in $\operatorname{QSYM}$. If we take an element $\chi(\mathbf{x})$ of
$C(\mathfrak{G},\operatorname{QSYM})$ and a given basis $B$ for quasisymmetric
functions, we say that $\chi(\mathbf{x})$ is _$B$ -effective_ if
$\chi(\mathbf{x})$ can expressed in the basis $B$ with coefficients that are
characters of representations of $\mathfrak{G}$. If we let
$\chi_{1},\ldots,\chi_{k}$ denote the irreducible characters of
$\mathfrak{G}$, then $R=\\{\chi_{i}B_{\alpha}:i\in[k],B_{\alpha}\in B\\}$
forms a basis for $C(\mathfrak{G},\operatorname{QSYM})$. If $\chi(\mathbf{x})$
is $B$-effective, then $\chi(\mathbf{x})$ can be expressed as a linear
combination of $\chi_{i}B_{\alpha}$ with nonnegative integer coefficients.

We prove that the $D$-partition quasisymmetric class function for locally
special posets is $F$-effective in Theorem 17. The notion of _locally special_
was first introduced by Grinberg, under the name tertispecial. He also
suggests locally special as an alternative name. Our results specialize to
both known and new results in the literature. We also give a proof that the
corresponding orbital $D$-partition enumerator is $F$-positive. This implies
that locally special double posets have $F$-positive $D$-partition
enumerators, which appears to be new. It also implies $F$-positivity for skew
Schur functions and for labeled $P$-partition enumerators.

We also prove in Theorem 19 that $\chi(G,\mathfrak{G},\mathbf{x})$ is also
$F$-effective.

We study polynomial invariants as well. There are lots of results about
inequalities for coefficients of chromatic polynomials of graphs with respect
to different bases, including recent work that the coefficients of
$(-1)^{n}\chi(G,-x)$ are unimodal [12] and strongly flawless [14]. Given a
sequence $(f_{0},\ldots,f_{d})$, we say the sequence is _strongly flawless_ if
the following inequalities are satisfied:

  1. (1)

for $0\leq i\leq\frac{d-1}{2}$, we have $f_{i}\leq f_{i+1}$.

  2. (2)

For $0\leq i\leq\frac{d}{2}$, we have $f_{i}\leq f_{d-i}$.

For this paper, we are focused on the sequence of coefficients for a
polynomial $p(x)$ with respect to the basis $\binom{x}{k}$. We refer to these
coefficients as the $f$-vector, and say $p(x)$ is _strongly flawless_ if the
$f$-vector is strongly flawless and nonnegative. We have a representation-
theoretic generalization: now the $f_{i}$ are required to be effective
characters, and we interpret inequalities of the form $f_{i}\leq f_{k}$ as
saying that $f_{k}-f_{i}$ is also a character. We refer to such a sequence of
characters as _effectively flawless_. We show that $\Omega(D,\mathfrak{G},x)$
and $\chi(G,\mathfrak{H},x)$ are effectively flawless in Section 6. Then we
obtain the following theorem:

We also discuss combinatorial reciprocity theorems. In [17], he defines a
combinatorial reciprocity theorem as ‘a result which establishes a kind of
duality between two enumeration problems’. The book by Beck and Sanyal [4] is
full of many examples of such results. In general, we suppose that we have a
vector subspace $V$ of a ring of formal power series, and that $V$ comes
equipped with an involution $\omega$. Given two generating functions $f,g\in
V$, a _combinatorial reciprocity theorem_ is the statement that $f=\omega g$.
This is more general than the examples that appear in Beck and Sanyal’s book,
but still fits the general notion Stanley originally proposed.

In this paper, $V$ is usually vector space of class functions from
$\mathfrak{G}$ to quasisymmetric functions of a fixed degree $d$, and
$\omega=(-1)^{d}S\operatorname{sgn}$. Hence a combinatorial reciprocity
theorem for a quasisymmetric class function consists of showing that
$(-1)^{d}S\operatorname{sgn}p(\mathbf{x})$ is $M$-realizable by giving an
explicit description of the resulting permutation characters. The
$\operatorname{sgn}$ term is the sign representation, which naturally arises
as $\mathfrak{G}$ is always a a permutation group. It arises naturally in the
work of Stapledon, Grinberg and Jochemko. We are also able to deduce
combinatorial reciprocity theorems for corresponding orbital invariants, and
for polynomial invariants as well.

We prove a combinatorial reciprocity theorem for double posets in Theorem 11,
which involves taking duals of partial orders, and a combinatorial reciprocity
theorem for digraphs in Theorem 14, which involves group actions on pairs
$(O,f)$, where $O$ is an acyclic orientation and $f$ is a compatible coloring.

The paper is organized as follows. In Section 2, we define quasisymmetric
functions, review some representation theory, and discuss set compositions. We
also discuss polynomials, and quasisymmetric class functions. In Section 3, we
define double posets, $D$-partitions and the corresponding $D$-partition
quasisymmetric class function. Then we prove some basic facts about
$\Omega(D,\mathfrak{G},\mathbf{x})$. We also discuss some properties about
locally special double posets that we need for later proofs. In Section 4, we
define the chromatic quasisymmetric class function, and provide a formula
expressing $\chi(G,\mathfrak{G},\mathbf{x})$ in terms of quasisymmetric class
functions related to double posets coming from acyclic orientations of $G$. In
Section 5, we prove our combinatorial reciprocity theorems for
$\Omega(D,\mathfrak{G},\mathbf{x})$ and $\chi(G,\mathfrak{G},\mathbf{x})$. In
Section 6, we show our polynomial invariants are effectively flawless. We also
show other properties about the quasisymmetric functions, and study some
examples to show how properties fail for $h$-vectors. In Section 7, we prove
$F$-effectiveness for $\Omega(D,\mathfrak{G},\mathbf{x})$ and
$\chi(G,\mathfrak{G},\mathbf{x})$. We also establish the corresponding
$h$-effectiveness for the related polynomial invariants, and deduce some
$F$-positivity results as corollaries. In Section 8, we define our orbital
quasisymmetric functions, and deduce facts about these invariants from the
results we have obtained about the quasisymmetric class functions. Finally, in
Section 9, we discuss some open problems.

##  2\. Preliminaries

Given a basis $B$ for a vector space $V$, and $\vec{\beta}\in B,\vec{v}\in V$,
we let $[\vec{\beta}]\vec{v}$ denote the coefficient of $\vec{\beta}$ when we
expand $\vec{v}$ in the basis $B$.

Let $\mathbf{x}=x_{1},x_{2},\ldots$ be a sequence of commuting indeterminates.
Let $n\in\mathbb{N}$ and let $f\in\mathbb{K}[[\mathbf{x}]]$ be a homogeneous
formal power series in $\mathbf{x}$, where the degree of every monomial in $f$
is $n$. Then $f$ is a _quasisymmetric function_ if it satisfies the following
property: for every $S=\\{i_{1},\ldots,i_{k}\\}$ with
$i_{1}<i_{2}<\cdots<i_{k}$, and every integer composition
$\alpha_{1}+\cdots+\alpha_{k}=n$, we have
$[\prod_{j=1}^{k}x_{i_{j}}^{\alpha_{j}}]f=[\prod_{j=1}^{k}x_{j}^{\alpha_{j}}]f$.
Often, we will define quasisymmetric functions that are generating functions
over functions. Given a function $f:S\to\mathbb{N}$, we define
$\mathbf{x}^{f}=\prod_{v\in S}x_{f(v)}$. For example, the chromatic symmetric
function of a graph $G$ is defined as
$\sum\limits_{f:V\to\mathbb{N}}\mathbf{x}^{f}$ where the sum is over all
proper colorings of $G$.

Given an integer composition
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})$ of $n$, we let

$M_{\alpha}=\sum\limits_{i_{1}<\cdots<i_{k}}\prod_{j=1}^{k}x_{i_{j}}^{\alpha_{j}}.$

These are the _monomial quasisymmetric functions_ , which form a basis for the
ring of quasisymmetric functions.

The second basis we focus on is Gessel’s basis of fundamental quasisymmetric
functions. The set of integer compositions is partially ordered by refinement.
With respect to this partial order, the set of integer compositions forms a
lattice. The _fundamental quasisymmetric function_ $F_{\alpha}$ are defined
by:

$F_{\alpha}=\sum\limits_{\beta\geq\alpha}M_{\beta}.$

There is a well-known bijection between subsets of $[n-1]$ of size $k-1$ and
integer compositions $\alpha\models n$ of length $k$, given by defining
$S(\alpha)=\\{\alpha_{1},\alpha_{1}+\alpha_{2},\ldots,\alpha_{1}+\alpha_{2}+\ldots+\alpha_{k-1}\\}$.
Under this bijection, the lattice of integer compositions is isomorphic to the
Boolean lattice. Given a set $A=\\{s_{1},\ldots,s_{k-1}\\}$ with
$s_{1}<s_{2}<\cdots<s_{k-1}$, we have
$S^{-1}(A)=(s_{1},s_{2}-s_{1},s_{3}-s_{2},\ldots,s_{k}-s_{k-1},n-s_{k})$.

There is an important linear transformation on quasisymmetric functions called
the _antipode_ :

$S(M_{\alpha})=(-1)^{\ell(\alpha)}\sum\limits_{\beta\leq\alpha}M_{\overleftarrow{\beta}}$

where $\overleftarrow{\beta}$ is the composition given by reversing the order
of $\beta$. Antipodes exist for any graded connected bialgebra, and are
analogous to inversion for groups.

Our proofs rely a lot on working with set compositions, and quasisymmetric
functions related to set compositions. Given a finite set $N$, a _set
composition_ is a sequence $(S_{1},\ldots,S_{k})$ of disjoint non-empty
subsets whose union is $N$. We denote set compositions as
$S_{1}|S_{2}|\cdots|S_{k}$, and refer to the sets $S_{i}$ as _blocks_. We use
$C\models N$ to denote that $C$ is a set composition of $N$, and let
$\ell(C)=k$ be the length of the composition. Given $C$, the associated
integer composition is $\alpha(C)=(|C_{1}|,|C_{2}|,\ldots,|C_{k}|)$. We refer
to $\alpha(C)$ as the _type_ of $C$. We partially order set compositions by
refinement. Finally given a set composition $C$ of type $\beta$ and
$\alpha\leq\beta$, let $C_{\alpha}(C)$ be the unique set composition of type
$\alpha$ such that $C_{\alpha}(C)\leq C$.

###  2.1. Group actions and class functions

Given a group action $\mathfrak{G}$ on a set $X$, we let $X/\mathfrak{G}$
denote the set of orbits. For $x\in X$, $\mathfrak{G}_{x}$ is the stabilizer
subgroup, and $\mathfrak{G}(x)$ is the orbit of $x$. Also, a _transversal_ is
a subset $T\subset X$ such that $|T\cap O|=1$ for every orbit $O$ of $X$.
Finally, for $\mathfrak{g}\in\mathfrak{G}$, we let
$\operatorname{Fix}_{\mathfrak{g}}(X)=\\{x\in X:\mathfrak{g}x=x\\}$.

There is an action of $\mathfrak{S}_{N}$ on the collection of all set
compositions of $N$. Given a permutation $\mathfrak{g}\in\mathfrak{S}_{N}$,
and a set composition $C\models N$, we let

$\mathfrak{g}C=\mathfrak{g}(C_{1})|\mathfrak{g}(C_{2})|\cdots|\mathfrak{g}(C_{k}).$

Then we obtain an action of $\mathfrak{S}_{N}$ on the collection of all set
compositions of $N$.

We assume familiarity with representation theory of finite groups - see [8]
for basic definitions. Recall that, given any group action of $\mathfrak{G}$
on a finite set $X$, there is a group action on $\mathbb{C}^{X}$ as well,
which gives rise to a representation. The resulting representations are called
_permutation representations_. We are working with representations over
$\mathbb{C}$. We let $C(\mathfrak{G})$ be the ring of class functions of
$\mathfrak{G}$. There is an orthonormal basis of $C(\mathfrak{G})$ given by
the characters of the irreducible representations of $\mathfrak{G}$. We refer
to elements $\chi\in C(\mathfrak{G})$ that are integer combinations of
characters as _virtual characters_ , and elements that are nonnegative integer
linear combinations as _effective characters_. Finally, we see $\chi$ is a
_permutation character_ if it is the character of a permutation
representation. We partially order $C(\mathfrak{G})$ by saying
$\chi\leq_{\mathfrak{G}}\psi$ if $\psi-\chi$ is an effective character.

Let $R$ be a $\mathbb{C}$-algebra. Let $C(\mathfrak{G},R)$ be the set of class
functions from $\mathfrak{G}$ to $R$. That is, for every
$\mathfrak{g},\mathfrak{h}\in\mathfrak{G}$, and $\chi\in C(\mathfrak{G},R)$,
we have $\chi(\mathfrak{hg}\mathfrak{h}^{-1})=\chi(\mathfrak{g}).$ For our
paper, $R$ is usually $\operatorname{QSYM}$ or $\mathbb{C}[x]$.

Let $\mathbf{B}$ be a basis for $R$. For $b\in\mathbf{B}$,
$\mathfrak{g}\in\mathfrak{G}$, and $\chi\in C(\mathfrak{G},R)$, let
$\chi_{b}(\mathfrak{g})=[b]\chi(\mathfrak{g})$. Then $\chi_{b}$ is also a
class function. Thus we can write
$\chi=\sum\limits_{b\in\mathbf{B}}\chi_{b}b$. Conversely, given a family
$\chi_{b}$ of class functions, one for each $b\in\mathbf{B}$ the function
$\chi$ defined by
$\chi(\mathfrak{g})=\sum\limits_{b\in\mathbf{B}}\chi_{b}(\mathfrak{g})b$ is a
class function in $C(\mathfrak{G},R)$. We say that $\chi$ if
$\mathbf{B}$-effective if $\chi_{b}$ is an effective character for all
$b\in\mathbf{B}$. We say that $\chi$ is B-realizable if $\chi_{b}$ is a
permutation character for all $b$. If $\mathbf{B}$ has a partial order on it,
then we say $\chi$ is $\mathbf{B}$-increasing if for all $b\leq c$ in
$\mathbf{B}$, we have $\chi_{b}\leq_{\mathfrak{G}}\chi_{c}$. Assuming $\chi$
is $\mathbf{B}$-effective, this is equivalent to saying that $\chi_{c}$ is the
character for the representation of $\mathfrak{G}$ on some module $V$, and
$\chi_{b}$ is the character of a representation of a submodule of $V$.

A _quasisymmetric class function_ is an element of
$C(\mathfrak{G},\operatorname{QSYM})$.

######  Proposition 1.

Let $\chi\in C(\mathfrak{G},\operatorname{QSYM})$ have degree $d$. If $\chi$
if $F$-effective, then $\chi$ is $M$-increasing.

###### Proof.

Let $\chi=\sum_{\alpha\models d}\psi_{\alpha}F_{\alpha}$. Then there exists
$\mathfrak{G}$-modules $W_{\alpha}$ such that $\psi_{\alpha}$ is the character
of the representation of $\mathfrak{G}$ on $W_{\alpha}$. If we let
$V_{\alpha}=\bigoplus_{\beta\leq\alpha}W_{\beta}$, Then $V_{\alpha}$ has
character $\sum_{\beta\leq\alpha}\psi_{\beta}=[M_{\alpha}]\chi$.

Let $\alpha\leq\beta$. Then we see that $V_{\alpha}$ is a submodule of
$V_{\beta}$. Hence $\chi_{\beta}-\chi_{\alpha}$ is the character of the
complement of $V_{\alpha}$ in $V_{\beta}$. Thus $\chi$ is $M$-increasing. ∎

Given a subgroup $\mathfrak{H}$ of $\mathfrak{G}$, and a class function
$\chi\in C(\mathfrak{H},R)$, We define the _induced class function_
$\chi\uparrow_{\mathfrak{H}}^{\mathfrak{G}}\in C(\mathfrak{G},R)$ by

$\chi\uparrow_{\mathfrak{H}}^{\mathfrak{G}}(\mathfrak{g})=\frac{1}{|\mathfrak{H}|}\sum\limits_{\mathfrak{k}\in\mathfrak{G}:\mathfrak{kg}\mathfrak{k}^{-1}\in\mathfrak{H}}\chi(\mathfrak{kg}\mathfrak{k}^{-1}).$

Finally, we define a function
$\langle\cdot,\cdot\rangle:C(\mathfrak{G},R)\times C(\mathfrak{G},R)\to R$ by
$\langle\chi,\psi\rangle=\frac{1}{|\mathfrak{G}|}\overline{\chi}(\mathfrak{g})\psi(\mathfrak{g})$
where $\overline{x}$ is the complex conjugate. In the case where
$R=\mathbb{C}$, this is the usual inner product on class functions.

######  Proposition 2.

Let $\mathfrak{G}$ be a finite group, let $R$ be a $\mathbb{C}$-algebra with
basis $\mathbf{B}$. Fix $\chi\in C(\mathfrak{G},R)$.

  1. (1)

For $b\in\mathbf{B}$, we have
$[b]\left(\chi\uparrow_{\mathfrak{H}}^{\mathfrak{G}}\right)=\left([b]\chi\right)\uparrow_{\mathfrak{H}}^{\mathfrak{G}}$.

  2. (2)

Given an irreducible character $\psi$, we have
$\langle\chi,\psi\rangle=\sum\limits_{b,c\in B}\langle\chi_{b},\psi_{c}\rangle
b\cdot c$.

  3. (3)

If $\chi$ is $\mathbf{B}$-effective, and $\psi$ is an irreducible character,
then $\langle\psi,\chi\rangle$ is $\mathbf{B}$-positive.

  4. (4)

Suppose B is partially ordered. Let $\psi\in C(\mathfrak{G})$. If $\chi$ is
$B$-increasing, then for all $b\leq c$ in B we have
$[b]\langle\psi,\chi\rangle\leq[c]\langle\psi,\chi\rangle.$

###### Proof.

Let $\mathfrak{g}\in\mathfrak{G}$. Then

$\displaystyle\chi\uparrow_{\mathfrak{H}}^{\mathfrak{G}}(\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{H}|}\sum\limits_{\mathfrak{k}\in\mathfrak{G}:\mathfrak{kg}\mathfrak{k}^{-1}\in\mathfrak{H}}\chi(\mathfrak{kg}\mathfrak{k}^{-1})$
$\displaystyle=\frac{1}{|\mathfrak{H}|}\sum\limits_{\mathfrak{k}\in\mathfrak{G}:\mathfrak{kg}\mathfrak{k}^{-1}\in\mathfrak{H}}\sum\limits_{b\in
B}\chi_{b}(\mathfrak{kg}\mathfrak{k}^{-1})b$ $\displaystyle=\sum\limits_{b\in
B}\left(\frac{1}{|\mathfrak{H}|}\sum\limits_{\mathfrak{k}\in\mathfrak{G}:\mathfrak{kg}\mathfrak{k}^{-1}\in\mathfrak{H}}\chi_{b}(\mathfrak{kg}\mathfrak{k}^{-1})\right)b$
$\displaystyle=\sum\limits_{b\in
B}\chi_{b}\uparrow_{\mathfrak{H}}^{\mathfrak{G}}(\mathfrak{g})b.$

Thus we see that the first result follows from comparing the coefficient of
$b$ on both sides.

Let $\psi\in C(\mathfrak{G},R)$. Then

$\displaystyle\langle\chi,\psi\rangle$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\bar{\chi}(\mathfrak{g})\psi(\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\left(\sum\limits_{b\in
B}\bar{\chi_{b}}(\mathfrak{g})b\right)\left(\sum\limits_{c\in
B}\psi_{c}(\mathfrak{g})c\right)$ $\displaystyle=\sum\limits_{b\in
B}\sum\limits_{c\in
B}\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\bar{\chi_{b}}(\mathfrak{g})\psi_{c}(\mathfrak{g})b\cdot
c.$

For the third claim, let $\psi$ be an irreducible character. Let
$b\in\mathbf{B}$. Using the second claim, we have
$[b]\langle\psi,\chi\rangle=\langle\psi,[b]\chi\rangle\geq 0.$ Hence
$\langle\psi,\chi\rangle$ is $B$-positive.

Now suppose that $\chi$ is B-increasing. Let $b\leq c\in\mathbf{B}$. Then
there is a representation of $\mathfrak{G}$ whose character is
$\rho:=[c]\chi-[b]\chi$. Using the second claim, we see that

$[c]\langle\psi,\chi\rangle-[b]\langle\psi,\chi\rangle=\langle\psi,[c]\chi-[b]\chi\rangle=\langle\psi,\rho\rangle\geq
0.$

Hence $[c]\langle\psi,\chi\rangle\geq[b]\langle\psi,\chi\rangle$. ∎

###  2.2. Principal specialization

Given a polynomial $p(x)$ of degree $d$, define $h(t)=(1-t)^{d+1}\sum_{m\geq
0}p(m)t^{m}$. The sequence of coefficients of $h(t)$ is the _$h$ -vector_ of
$p(x)$. We define the $f$-vector $(f_{0},\cdot,f_{d})$ via
$p(x)=\sum_{i=0}^{d}f_{i}\binom{x}{i}$. We say that $p(x)$ is _strongly
flawless_ if the following inequalities are satisfied:

  1. (1)

for $0\leq i\leq\frac{d-1}{2}$, we have $f_{i}\leq f_{i+1}$.

  2. (2)

For $0\leq i\leq\frac{d}{2}$, we have $f_{i}\leq f_{d-i}$.

There is a lot of interest in log-concave and unimodal sequences in
combinatorics. We consider strongly flawless sequences to also be interesting,
as strongly flawless unimodal sequences can be seen as a generalization of
symmetric unimodal sequences. Examples of results with strongly flawless
sequences include the work of Hibi [11] and Juhnke-Kubitzke and Van Le [14].

Given a quasisymmetric function $F(\mathbf{x})$ of degree $d$, there is an
associated polynomial $\operatorname{ps}(F)(x)$ given by principal
specialization. For $x\in\mathbb{N}$, we set

$x_{i}=\begin{cases}1&i\leq x\\\ 0&i>x\end{cases}$

The resulting sequence is a polynomial function in $x$ of degree $d$, which we
denote by $ps(F)(x)$. If we write $F(\mathbf{x})=\sum\limits_{\alpha\models
d}c_{\alpha}M_{\alpha}$, then $f_{i}=\sum\limits_{\alpha\models
d:\ell(\alpha)=i}c_{\alpha}$. Similarly, if we write
$F(\mathbf{x})=\sum\limits_{\alpha\models n}d_{\alpha}F_{\alpha}$, then
$h_{i}=\sum\limits_{\alpha\models d:\ell(\alpha)=i}d_{\alpha}$.

The set $\mathbb{K}[x]$ is a Hopf algebra, with antipode given by
$Sp(x)=p(-x)$. Also, $\varphi:\operatorname{QSYM}\to\mathbb{K}[x]$ given by
$\varphi(F(\mathbf{x}))=\operatorname{ps}(F)(x)$ is a Hopf algebra
homomorphism and $\varphi(SF(\mathbf{x}))=f(-x)$.

Let $F(\mathbf{x})\in\operatorname{QSYM}$ be of degree $n$, and
$\mathfrak{g}\in\mathfrak{G}$. Define $\operatorname{ps}(F)\in
C(\mathfrak{G},\mathbb{C}[x])$ by
$\operatorname{ps}(F)(\mathfrak{g})=\operatorname{ps}(F(\mathbf{x};\mathfrak{g})).$
Then $\operatorname{ps}(F)$ is also the principal specialization, resulting in
an polynomial class function. If we write
$\operatorname{ps}(F)=\sum\limits_{i=0}^{d}f_{i}\binom{x}{i}$, then
$(f_{0},\ldots,f_{d})$ is the _equivariant $f$-vector_ of
$\operatorname{ps}(F)$, which consists of permutation characters. If we write
$\sum\limits_{m\geq 0}\operatorname{ps}(F)t^{m}=\frac{h(t)}{(1-t)^{n}}$, then
the coefficients of $h(t)$ is the _equivariant $h$-vector_ of
$\operatorname{ps}(F)$. Note that the entries of the equivariant $h$-vector
are virtual characters. We say $\operatorname{ps}(F)$ is $h$-effective if the
entries are effective characters. We say that $\operatorname{ps}(F)$ is
_effectively flawless_ if we have the following system of inequalities:

  1. (1)

for $0\leq i\leq\frac{d-1}{2}$, we have $f_{i}\leq_{\mathfrak{G}}f_{i+1}$.

  2. (2)

For $0\leq i\leq\frac{d}{2}$, we have $f_{i}\leq_{\mathfrak{G}}f_{d-i}$.

We can obtain results about polynomial class functions from the corresponding
quasisymmetric class functions.

######  Proposition 3.

Let $F(\mathbf{x})\in\operatorname{QSYM}$ be of degree $d$, and
$\mathfrak{g}\in\mathfrak{G}$.

  1. (1)

If we write $F(\mathbf{x})=\sum\limits_{\alpha\models
d}\chi_{\alpha}M_{\alpha}$, then
$\operatorname{ps}(F)=\sum\limits_{i=0}^{d}\chi_{\alpha}\binom{x}{\ell(\alpha)}$.

  2. (2)

If we write $F(\mathbf{x})=\sum\limits_{\alpha\models
d}\psi_{\alpha}F_{\alpha}$, then $h(t)=\sum\limits_{\alpha\models
d}\psi_{\alpha}t^{\ell(\alpha)}$. If $F(\mathbf{x})$ if $F$-effective, then
$\operatorname{ps}(F)$ is $h$-effective.

  3. (3)

If $G(\mathbf{x})\in C(\mathfrak{G},\operatorname{QSYM})$ with
$(-1)^{d}\operatorname{sgn}SF(\mathbf{x})=G(\mathbf{x}),$ then

$(-1)^{d}\operatorname{sgn}\operatorname{ps}(F)(-x)=\operatorname{ps}(G)(x).$

  4. (4)

If $F(\mathbf{x})$ is $M$-realizable and $M$-increasing, then
$\operatorname{ps}(F)$ is effectively flawless.

  5. (5)

Let $\psi$ be an irreducible character. If $F(\mathbf{x})$ is $F$-effective,
then $\langle\psi,\operatorname{ps}(F)\rangle$ is $h$-positive. If
$F(\mathbf{x})$ is $M$-increasing, then
$\langle\psi,\operatorname{ps}(F)\rangle$ is strongly flawless.

###### Proof.

The first three results are proven in a similar manner. Let
$\mathfrak{g}\in\mathfrak{G}$. Then
$\operatorname{ps}(F)(x;\mathfrak{g})=\operatorname{ps}(F(\mathbf{x};\mathfrak{g}))(x)$.
Since $F(\mathbf{x},\mathfrak{g})=\sum\limits_{\alpha\models
d}\chi_{\alpha}(\mathfrak{g})M_{\alpha}$, we have

$\operatorname{ps}(F(\mathbf{x};\mathfrak{g}))=\sum\limits_{\alpha\models
d}\chi_{\alpha}(\mathfrak{g})\binom{x}{\ell(\alpha)}.$

The result follows.

For the fourth result, let $d$ be the degree of $F(\mathbf{x})$. For each
$\alpha\models d$, let $V_{\alpha}$ be a $\mathfrak{G}$-module with character
$[M_{\alpha}]F(\mathbf{x})$. Since $F(\mathbf{x})$ is $M$-increasing, we know
there exists injective $\mathfrak{G}$-invariant functions
$\theta_{\alpha,\beta}:V_{\alpha}\to V_{\beta}$ for every
$\alpha\leq\beta\models d$. We let $V_{i}=\bigoplus\limits_{\alpha\models
d:\ell(\alpha)=i}V_{\alpha}$. Then the character of $V_{i}$ is $f_{i}$. To
show the inequalities, it suffices to find $\mathfrak{G}$-invariant injections
between $V_{i}$ and $V_{j}$. Then $f_{j}-f_{i}$ is the character of the
complement of $V_{i}$ in $V_{j}$.

We need to recall that the boolean lattice, and hence the lattice of integer
compositions, has a symmetric chain decomposition, a result due to DeBruijn
[6]. Let $C(d)$ be the set of integer compositions of $d$. A symmetric chain
decomposition is a partition of $C(d)$ into saturated chains
$c_{1},\ldots,c_{m}$ with the property that, for each chain $c_{i}$, the sum
of the ranks of the first and last element of $c_{i}$ is $d$.

Fix a symmetric chain decomposition $D$. Fix integers $i$ and $j$ such that
$1\leq i<j\leq d-i$. Consider an integer composition $\alpha$ with
$\ell(\alpha)=i$. Then there exists a chain $x_{1}<x_{2}<\cdots<x_{k}$ in $D$
with $x_{i-\ell(x_{1})+1}=\alpha$. Define
$\varphi_{i,j}(\alpha)=x_{j-\ell(x_{1})+1}$. We see that the following two
facts are true:

  1. (1)

If $i\leq\frac{d-1}{2}$, then $\varphi_{i,i+1}$ is injective.

  2. (2)

If $i<\frac{d}{2}$, then $\varphi_{i,d-i}$ is a bijection.

Let $1\leq i\leq\frac{d-1}{2}$, and let $\alpha\models d$ with
$\ell(\alpha)=i$. We define $\theta_{i,j}:V_{i}\to V_{j}$ by requiring
$\theta_{i,j}|_{V_{\alpha}}=\theta_{\alpha,\varphi_{i,j}(\alpha)}$. Then
$\theta_{i,j}(V_{\alpha})\subseteq V_{\varphi_{i,j}(\alpha)}$.

Thus $\theta_{i,j}$ is an injective $\mathfrak{G}$-invariant map. Thus $V_{i}$
is isomorphic to a submodule of $V_{i+1}$, and we have
$f_{i}\leq_{\mathfrak{G}}f_{i+1}$.

Now let $i\leq\frac{d}{2}$. Let $\theta_{i}:V_{i}\to V_{d-i}$ be given by
$\theta_{i}|_{V_{\alpha}}=\theta_{\alpha,\varphi_{i,d-i}(\alpha)}$. By a
similar argument, $\theta_{i}$ is injective and $\mathfrak{G}$-invariant.
Hence $V_{i}$ is isomorphic to a submodule of $V_{d-i}$, and
$f_{i}\leq_{\mathfrak{G}}f_{d-i}$.

For the last result, let $\psi$ be an irreducible character. A simple
calculation shows that
$\langle\psi,\operatorname{ps}(F)\rangle=\operatorname{ps}(\langle\psi,F\rangle)$.
If $F(\mathbf{x})$ is $F$-effective, then $\langle\psi,F\rangle$ is
$F$-positive. Since the entries of the $h$-vector are non-negative sums of
coefficients in the $F$-basis, the $h$-vector of
$\operatorname{ps}(\langle\psi,F,\rangle)$ is non-negative.

Finally, suppose that $F(\mathbf{x})$ is $M$-increasing. Viewing
$\langle\psi,F(\mathbf{x})\rangle$ as a quasisymmetric class function for the
trivial group. Then $\langle\psi,F\rangle$ is $M$-increasing. Hence
$\operatorname{ps}(\langle\psi,F\rangle)$ is effectively flawless. Since we
are working with the trivial group, we conclude that
$\langle\psi,\operatorname{ps}(F)\rangle$ is strongly flawless.

∎

##  3\. Double Posets

Now we will discuss double posets. The Hopf algebra of double posets was
introduced by Malvenuto and Reutenauer [15]. Grinberg associated a
quasisymmetric function to any double poset, which is a generalization of
Gessel’s $P$-partition enumerator. This quasisymmetric function is studied
extensively by Grinberg [10], who proved a combinatorial reciprocity theorem.

Given a finite set $N$, a _double poset_ on $N$ is a triple
$(N,\leq_{1},\leq_{2})$ where $\leq_{1}$ and $\leq_{2}$ are both partial
orders on $N$. Often for standard poset terminology, we will use $\leq_{i}$ as
a prefix to specify which of the two partial orders is being referred to. For
instance, a $\leq_{1}$-order ideal is a subset that is an order ideal with
respect to the first partial order, and a $\leq_{1}$-covering relation refers
to a pair $(x,y)$ such that $x\prec_{1}y$.

$b$$a$$c$$d$ | $c$$b$$d$$a$
---|---
Figure 1. A double poset.

Let $D$ be a double poset on a finite set $N$, and let $f:N\to\mathbb{N}$.
Then $f$ is a $D$-partition if and only if it satisfies the following two
properties:

  1. (1)

For $i\leq_{1}j$ in $D$, we have $f(i)\leq f(j)$.

  2. (2)

For $i\leq_{1}j$ and $j\leq_{2}i$ in $D$, we have $f(i)<f(j)$.

Let $P_{D}$ be the set of $D$-partitions. We define the $D$-partition
enumerator by

(1) $\Omega_{D,\varphi}(D,\mathbf{x})=\sum\limits_{f\in P_{D}}\prod_{v\in
N}x_{f(v)}$

This quasisymmetric function is studied extensively by Grinberg [10].

Given a double poset $D$, a pair $(m,m^{\prime})\in M$ is an _inversion_ if
$m<_{1}m^{\prime}$ and $m^{\prime}<_{2}m$. Given a set composition $C\models
N$, we say that $C$ is a $D$-set composition if it satisfies the following two
properties:

  1. (1)

For every $i$, $C_{1}\cup C_{2}\cup\cdots\cup C_{i}$ is a $\leq_{1}$-order
ideal.

  2. (2)

For every $i$, there are no inversions in $C_{i}$

Let $X_{D}$ be the set of $D$-set compositions.

######  Proposition 4.

Let $D$ be a double poset. Then $\Omega(D,\mathbf{x})=\sum\limits_{C\in
X_{D}}M_{\alpha(C)}.$

Given a double poset $D$, an automorphism is a bijection $\sigma:N\to N$ such
that, for all $x,y\in N$ and all $i\in\\{1,2\\}$, we have $x\leq_{i}y$ if and
only if $\sigma(x)\leq_{i}\sigma(y)$. We let $\operatorname{Aut}(D)$ be the
automorphism group of $D$. For instance, for the double poset in Figure 1, the
permutation $(ac)(bd)$ is the only nontrivial automorphism. Similarly the only
nontrivial automorphism of the double poset in Figure 2 is the permutation
$(a)(bd)(c)$.

Let $\mathfrak{G}\subseteq\operatorname{Aut}(D)$. For
$\mathfrak{g}\in\mathfrak{G}$, define

$\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(P_{D})}\mathbf{x}^{f}.$

This is the _$D$ -partition quasisymmetric class function_.

Naturally, there is an _order polynomial class function_ as well: given a
positive integer $n$, we let $X_{n,D}$ be the set of $D$-partitions
$\sigma:D\to[n]$. Then $\mathfrak{G}$ acts on $X_{n,D}$ and we let
$\Omega(D,\mathfrak{G},n)$ be the resulting character.

We give two alternative formulas for
$\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})$, and another formula for the
order polynomial class function. Let $X_{\alpha,D}$ be the set of $D$-set
compositions of type $\alpha$. Then $\mathfrak{G}$ acts on $X_{\alpha,D}$. Let
$\chi_{\alpha,D}$ be the resulting character.

######  Theorem 5.

Let $D$ be a double poset on a finite set $N$ and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Then we have the following
identities:

  1. (1)

$\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum\limits_{C\in\operatorname{Fix}_{\mathfrak{g}}(X_{D})}M_{\alpha(C)}$

  2. (2)

$\Omega(D,\mathfrak{G},\mathbf{x})=\sum\limits_{\alpha\models|N|}\chi_{\alpha,D}M_{\alpha}$

  3. (3)

$\Omega(D,\mathfrak{G},x)=\sum\limits_{\alpha\models|N|}\chi_{\alpha,D}(D,\mathfrak{G})\binom{x}{|\alpha|}.$

abcd | $b$$a$$c$$d$
---|---
Figure 2. A double poset.

###### Proof.

Fix a double poset $D$ on a finite set $N.$

For the first formula, let $f\in\operatorname{Fix}_{\mathfrak{g}}(P_{D})$. Let
$i_{1}<i_{2}<\cdots<i_{k}$ be the natural numbers for which
$f^{-1}(i_{j})\neq\emptyset$. Define
$C(f)=f^{-1}(i_{1})|f^{-1}(i_{2})|\cdots|f^{-1}(i_{k})$. This is the
composition associated with $f$. We see that

$M_{\alpha(C)}=\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(P_{D})}\mathbf{x}^{f}$

and that $\mathfrak{g}f=f$ if and only if $\mathfrak{g}C(f)=C(f)$. Thus we
obtain

$\sum\limits_{C\in\operatorname{Fix}_{\mathfrak{g}}(X_{D})}M_{\alpha(C)}=\sum\limits_{C\in\operatorname{Fix}_{\mathfrak{g}}(X_{D})}\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(P_{D})}\mathbf{x}^{f}=\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(P_{D})}\mathbf{x}^{f}.$

Given $\mathfrak{g}$, we have

$\sum\limits_{C\in\operatorname{Fix}_{\mathfrak{g}}(X_{D})}M_{\alpha(C)}=\sum\limits_{\alpha\models|N|}\sum\limits_{C\in\operatorname{Fix}_{\mathfrak{g}}(X_{\alpha,D})}M_{\alpha}=\sum\limits_{\alpha\models|N|}\chi_{\alpha,D}(\mathfrak{g})M_{\alpha}$

and so by the first formula, we conclude that
$\Omega(D,\mathfrak{G},\mathbf{x})=\sum\limits_{\alpha\models|N|}\chi_{\alpha,D}M_{\alpha}$.

For the third formula, let $\mathfrak{G}$ act trivially on $\binom{[n]}{k}$,
the collection of $k$-subsets of $[n]$. Then for any integer composition
$\alpha\models|N|$, we have an action of $\mathfrak{G}$ on
$\binom{[n]}{\ell(\alpha)}\times X_{\alpha,D}$. Given a $D$-partition
$f:D\to[n]$, let $\varphi(f)=(f(N),C(f))$. Then this defines an isomorphism of
$\mathfrak{G}$-sets:

$X_{n,D}\simeq\bigcup_{\alpha\models|N|}\binom{[n]}{\ell(\alpha)}\times
X_{\alpha,D}$

The result follows from taking linear spans to obtain $\mathfrak{G}$-modules,
and then taking the trace to obtain characters. ∎

As an example, consider the double poset $D$ in Figure 1, and let
$\mathfrak{G}=\operatorname{Aut}(D)$. Let $\rho$ denote the regular
representation. Then

$\displaystyle\Omega(D,\mathfrak{G},\mathbf{x})$
$\displaystyle=M_{2,2}+\rho(M_{1,1,2}+M_{1,1,1,1})$
$\displaystyle=F_{2,2}+\operatorname{sgn}(F_{1,1,2}+F_{2,1,1}-F_{1,1,1,1})+\rho
F_{1,2,1}.$

As another example, consider the double poset $D$ in Figure 2, and let
$\mathfrak{G}=\operatorname{Aut}(D)$. Let $\rho$ denote the regular
representation. Then

$\displaystyle\Omega(D,\mathfrak{G},\mathbf{x})$
$\displaystyle=M_{1,3}+M_{1,2,1}+\rho(M_{1,1,2}+M_{1,1,1,1})$
$\displaystyle=F_{1,3}+\operatorname{sgn}F_{1,1,2}.$

###  3.1. Properties of Double Posets

A double poset is _locally special_ if whenever $y$ $\leq_{1}$-covers $x$,
then $x$ and $y$ are $\leq_{2}$-comparable. The double poset in Figure 2 is
locally special, while the double poset in Figure 1 is not. Grinberg gives
several examples of locally special posets, including double posets coming
from skew shapes and labeled posets.

We say that an inversion pair $(x,y)$ is a _descent pair_ if $x\prec_{1}y$.

######  Lemma 6.

Let $D$ be a locally special double poset, and let $I\subseteq J$ be
$\leq_{1}$-order ideals. If $D$ has an inversion pair $(x,y)$ with $x,y\in
J\setminus I$, then $D$ has a descent pair $(w,z)$ with $w,z\in J\setminus I$.

###### Proof.

We prove the result by induction on $|J\setminus I|$. Let
$x\prec_{1}t\leq_{1}z$. Since $D$ is locally special, we have $x\leq_{2}t$ or
$t\leq_{2}x$. In the latter case, we have found a descent pair $(x,t)$. In the
former case the pair $(t,y)$ forms an inversion pair. We observe that then
interval $[t,y]$ is equal to $(y)\setminus(t)$, where $(a)$ is the principal
order ideal generated by $a$. Since $|[t,y]|<|J\setminus I|$, by induction
there is a descent pair $(w,z)$ in $[t,y]$, and we have
$x\leq_{1}\prec_{1}z\leq y$. ∎

For any set $S\subseteq N$, and a partial order $P$ on $N$, we let $P|_{S}$
denote the induced poset on $S$. The same notation is also used for linear
orders (which are a special case of partial orders), and for double posets.

We say that a linear order $\ell$ of $N$ is $D$-compatible if only if for all
pairs $(I,J)$ of $\leq_{1}$-order ideals with $I\subseteq J$, the linear order
$\ell|_{J\setminus I}$ is a $\leq_{1}$-linear extension of $D|_{J\setminus I}$
if and only if $D|_{J\setminus I}$ contains no inversion pairs.

######  Lemma 7.

Let $D$ be a double poset on a finite set $N$. If $D$ is locally special, then
there exists a $D$-compatible linear order.

###### Proof.

Let $D$ be a locally special double poset. Let $G(D)$ be the directed graph
obtained by taking the directed edges of the Hasse diagram of $\leq_{1}$, and
reversing the direction on edges $x\prec_{1}y$ if $x>_{2}y$. We claim that
$G(D)$ is acyclic. Suppose that we have a directed cycle $C$ in $G(D)$. Let
$C$ have vertices $x_{0},x_{1},\ldots,x_{k}$ in order. Note that this means
that $x_{i}\prec_{1}x_{i+1}$ or $x_{i+1}\prec_{1}x_{i}$ for all $i$. Since $D$
is locally special, we have $x_{i}\leq_{2}x_{i+1}$ for all $i$, which is a
contradiction. Thus there is no directed cycle.

We say $x\leq_{P}y$ if there is a directed path from $y$ to $x$ in $G(D).$ Let
$\ell$ be a linear extension of $P$. We claim that $\ell$ is $D$-compatible.

Let $I\subseteq J$ be $\leq_{1}$-order ideals. Suppose that there are no
inversions in $J\setminus I$. Then we see that $P|_{J\setminus
I}=D|_{J\setminus I}$, and thus $\ell|_{J\setminus I}$ is a linear extension
of $D|_{J\setminus I}$. Suppose instead there is an inversion pair $(x,y)$ in
$J\setminus I$. By Lemma 6, we can choose $(x,y)$ to be a descent pair. Since
$x\geq_{2}y$, we have $(x,y)$ is a directed edge in $G(D)$, and thus
$x\geq_{P}y$. Since $\ell$ is a linear extension of $P$, we have $x>_{\ell}y$.
Therefore $\ell|_{J\setminus I}$ is not a linear extension of $D|_{J\setminus
I}$.

∎

Given a $D$-compatible linear order $\ell$, we can lexicographically order any
other linear order of $N$: given two linear orders $\pi$ and $\sigma$,
consider the first $i$ where $\pi_{i}\neq\sigma_{i}$. Then we say
$\pi<_{\ell}\sigma$ if $\pi_{i}<_{\ell}\sigma$. Let $C$ be a $D$-set
composition for $D$. We let $\ell(C)$ be the lexicographically first total
refinement of $C$. Finally, we can also totally preorder $X_{\alpha,D}$, the
set of $D$-set compositions of type $\alpha$. Given $C,C^{\prime}\in
X_{\alpha,D}$, we say $C\leq_{\ell}C^{\prime}$ if and only if
$\ell(C)\leq_{\ell}\ell(C^{\prime})$. Note that it is a preorder because it is
possible for $\ell(C)=\ell(C^{\prime})$.

We see that a $D$-set composition with only singleton blocks is a
$\leq_{1}$-linear extension. We prove a proposition regarding when such linear
extensions are increasing with respect to $\ell$. We say that a
$\leq_{1}$-linear extension $\pi$ is _increasing_ if
$\pi_{1}<_{\ell}\pi_{2}<_{\ell}\cdots<_{\ell}\pi_{n}$.

######  Proposition 8.

Let $D$ be a locally special double poset, and let $\ell$ be a $D$-compatible
linear order. Let $I\subseteq J$ be $\leq_{1}$-order ideals of $D$. Then
$D|_{J\setminus I}$ has an increasing $\leq_{1}$-linear extension $\sigma$ if
and only if $D$ has no inversions in $J\setminus I$. In that case,
$\ell|_{J\setminus I}=\sigma$, and $\sigma$ is lexicographically least.

###### Proof.

We prove the result by induction on $k=|J\setminus I|.$ Suppose that
$D|_{J\setminus I}$ has an increasing $\leq_{1}$-linear extension $\sigma$.
Then $D|_{J\setminus(I\cup\\{\sigma_{1}\\})}$ also has a $\leq_{1}$-increasing
linear extension. By induction, we see that
$\sigma|_{J\setminus\\{\sigma_{1}\\}}=\ell|_{J\setminus
I\cup\\{\sigma_{1}\\}}$. Similarly, if we let
$J^{\prime}=J\setminus\\{\sigma_{k}\\}$ and we see that
$D|_{J^{\prime}\setminus I}$ has an increasing $\leq_{1}$-linear extension,
and thus
$\sigma|_{\\{\sigma_{1},\ldots,\sigma_{k-1}\\}}=\ell|_{J^{\prime}\setminus
I}$. Therefore, we have $\ell|_{J\setminus I}=\sigma$. Hence
$\ell|_{J\setminus I}$ is a linear extension of $D|_{J\setminus I}$, and by
definition of $D$-compatible order, this means that $J\setminus I$ does not
contain any inversions.

Now we suppose that $J\setminus I$ has no inversions. Then $\ell$ restricted
to $J\setminus I$ is a linear extension of $D|_{J\setminus I}$ with respect to
$\leq_{1}$. Moreover, $\ell|_{J\setminus I}$ is increasing.

Let $\sigma=\ell|_{J\setminus I}$. Let $\tau$ be another increasing
$\leq_{1}$-linear extension. Suppose $\tau_{1}\neq\sigma_{1}$. Then
$\tau_{k}=\sigma_{1}$ for some $k>1$. However, then
$\tau_{1}>_{\ell}\tau_{k}$, and hence $\tau$ is not increasing. Thus
$\tau_{1}=\sigma_{1}$. By induction, we have
$\tau|_{J\setminus\\{\tau_{1}\\}}$ and $\sigma|_{J\setminus\\{\sigma_{1}\\}}$
are both increasing $\leq_{1}$-linear extensions of
$D|_{J\setminus(I\cup\\{\tau_{1}\\})}$, and hence are equal by induction. Thus
$\sigma=\tau$.

∎

##  4\. Digraph coloring

We refer to directed graphs as digraphs. We require that there is at most one
directed edge between any two vertices. An example appears in Figure 3.

ABCD Figure 3. A digraph

Given a digraph $G$ on $N$, a _coloring_ is a function $f:N\to\mathbb{N}$
which satisfies:

  1. (1)

for every edge $(u,v)$, $f(u)\neq f(v)$.

An edge $(u,v)$ is an $f$-ascent if $f(u)<f(v)$. We let
$\operatorname{asc}(f)$ denote the number of $f$-descents. An example of a
coloring in Figure 3 is given by $f(A)=1$, $f(B)=2$, $f(C)=3$, and $f(D)=4$.
This coloring has exactly one ascent, from $D$ to $A$. Let $C_{G}$ denote the
set of all colorings, and we let $C_{n,G}=\\{f\in C_{G}:f(N)\subseteq[n]\\}.$
Finally, we let $C_{k,n,G}=\\{f\in C_{n,G}:\operatorname{asc}(f)=k\\}$.

######  Definition 9.

The _chromatic quasisymmetric function_ is

$\chi(G,,\mathbf{x})=\sum\limits_{f\in
C_{G}}t^{\operatorname{asc}(f)}\mathbf{x}^{f}.$

Likewise, for $n\in\mathbb{N}$, define the _chromatic polynomial_ to be

$\chi(G,n)=\sum\limits_{f\in C_{n,G}}t^{\operatorname{asc}(f)},$

where we sum over all proper colorings $f:I\to[n]$.

For example, for the digraph $G$ in Figure 3, we have
$\chi(G,\mathbf{x})=2t^{2}M_{2,2}+4t^{2}M_{2,1,1}+4t^{2}M_{1,2,1}+4t^{2}M_{1,1,2}+(4t^{3}+16t^{2}+4t)M_{1,1,1,1}.$
For instance, $2M_{2,2}$ comes from colorings $f$ where $f(A)=f(C)$ and
$f(B)=f(D)$. In all such cases, there ends up being two ascents.

Now we define the automorphism group of a digraph. Given a digraph $G$ on a
finite set $N$, a bijection $\mathfrak{g}:N\to N$ is an _automorphism_ if for
every $u,v\in N$, we have $(u,v)\in E(G)$ if and only if
$(\mathfrak{g}(u),\mathfrak{g}(v))\in E(G)$. Let $\operatorname{Aut}(G)$ be
the set of automorphisms of $G$, which forms a group. For the digraph $G$
appearing in Figure 3, the automorphism group is isomorphic to $C_{4}$, the
cyclic group of order $4$, acting by rotations.

Now we define the chromatic quasisymmetric class function. Let
$\mathfrak{G}\subset\operatorname{Aut}(G)$. For $\mathfrak{g}\in\mathfrak{G}$,
we let

$\chi(G,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(C_{G})}t^{\operatorname{asc}(f)}\mathbf{x}^{f}$

where the sum is over all proper colorings of $G$. This defines a class
function on $\mathfrak{G}$ whose values are quasisymmetric functions over
$\mathbb{Q}[t]$. This is the _chromatic quasisymmetric class function_
associated to $G$.

As an example, consider the digraph in Figure 3, and let
$\mathfrak{G}=\mathbb{Z}/4\mathbb{Z}$ act via rotation. Let $\rho$ denote the
regular representation. Then

$\chi(G,\mathfrak{G},t,\mathbf{x})=(1+\operatorname{sgn})t^{2}M_{2,2}+\rho
t(tM_{2,1,1}+tM_{1,2,1}+tM_{1,1,2})+(1+4t+t^{2})M_{1,1,1,1}.$

We let $\chi_{i}:\mathbb{Z}/4\mathbb{Z}\to\mathbb{C}$ be given by
$\chi_{i}(j)=i^{j}$, and $\chi_{-i}:\mathbb{Z}/4\mathbb{Z}\to\mathbb{C}$ be
given by $\chi_{-i}(j)=(-i)^{j}$. Then

$\displaystyle\chi(G,\mathfrak{G},t,\mathbf{x})$
$\displaystyle=(1+\operatorname{sgn})t^{2}(F_{2,2}+3F_{1,1,1,1})+t^{2}(\chi_{i}+\chi_{-i})(F_{2,1,1}+F_{1,1,2})$
$\displaystyle+\rho(tF_{1,1,1,1}+t^{2}F_{1,2,1}+t^{3}F_{1,1,1,1}).$

We detail some formulas relating the chromatic quasisymmetric class function
of a digraph $G$ to $D$-partition quasisymmetric class functions. The key
concept for proving identities is an acyclic orientation. For a directed graph
$G$, an acyclic orientation is another digraph $O$ on the same vertex set,
with no directed cycles, such that, for every $u,v\in N$, we have $(u,v)\in G$
if and only if $(u,v)\in E(O)$ or $(v,u)\in E(O)$. An $O$-ascent is an edge
$(u,v)\in E(D)$ where $(u,v)\in E(O)$. We let $\operatorname{asc}(O)$ be the
number of $O$-ascents. We let $\mathcal{A}(G)$ be the set of acyclic
orientations. Given an acyclic orientation $O$, there is a natural double
poset associated with $O$: for $x,y\in N$, we say $x\leq_{1}y$ if and only if
there is a directed path from $y$ to $x$ in $O$. Then we define $x\leq_{2}y$
if and only if $y\leq_{1}x$.

######  Lemma 10.

Let $G$ be a directed graph and let
$\mathfrak{G}\subseteq\operatorname{Aut}(G)$.

  1. (1)

For $\mathfrak{g}\in\mathfrak{G}$, we have

$\chi(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g})=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(\mathcal{A}(G))}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{g}).$

  2. (2)

We have

$\chi(G,\mathfrak{G},t,\mathbf{x})=\sum\limits_{O\in\mathcal{A}(G)}\frac{t^{\operatorname{asc}(O)}}{|\mathfrak{G}(O)|}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x})\uparrow_{\mathfrak{G}_{O}}^{\mathfrak{G}}.$

  3. (3)

Let $\mathcal{T}$ be a transversal for the group action of $\mathfrak{G}$ on
$A(G)$. We have

$\chi(G,\mathfrak{G},\mathbf{x})=\sum\limits_{O\in\mathcal{T}}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x})\uparrow_{\mathfrak{G}_{O}}^{\mathfrak{G}}.$

###### Proof.

Let $\mathfrak{g}\in\mathfrak{G}$, and let $f$ be a proper coloring of $G$.
Let $O_{f}$ be the orientation of $G$ given by directing $v$ to $u$ if
$f(v)>f(u)$. Then $O_{f}$ is an acyclic orientation. We also see that
$\mathfrak{g}O_{f}=O_{f}$, and that
$\operatorname{asc}(f)=\operatorname{asc}(O_{f}).$ Then $f$ is a
$P_{O_{f}}$-partition. Thus

$\displaystyle\chi(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(C_{G})}t^{\operatorname{asc}(f)}\mathbf{x}^{f}$
$\displaystyle=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(A(G))}t^{\operatorname{asc}(O)}\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(C_{O}):O_{f}=O}\mathbf{x}^{f}$
$\displaystyle=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(A(G))}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{g}).$

To prove our second formula, we have

$\displaystyle\sum\limits_{O\in
A(G)}\frac{t^{\operatorname{asc}(O)}}{|\mathfrak{G}(O)|}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x})\uparrow_{\mathfrak{G}_{O}}^{\mathfrak{G}}(\mathfrak{g})$
$\displaystyle=\sum\limits_{O\in
A(G)}\frac{t^{\operatorname{asc}(O)}}{|\mathfrak{G}(O)|}\frac{1}{|\mathfrak{G}_{O}|}\sum\limits_{\mathfrak{h}\in\mathfrak{G}:\mathfrak{hg}\mathfrak{h}^{-1}\in\mathfrak{G}_{O}}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{hg}\mathfrak{h}^{-1})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{h}\in\mathfrak{G}}\sum\limits_{\begin{subarray}{c}O\in
A(G)\\\
\mathfrak{hg}\mathfrak{h}^{-1}\in\mathfrak{G}_{O}\end{subarray}}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{hg}\mathfrak{h}^{-1})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{h}\in\mathfrak{G}}\sum\limits_{\begin{subarray}{c}O\in
A(G)\\\
\mathfrak{g}\in\mathfrak{G}_{\mathfrak{h}^{-1}O}\end{subarray}}t^{\operatorname{asc}(O)}\Omega(P_{\mathfrak{h}^{-1}O},\mathfrak{G}_{\mathfrak{h}^{-1}O},\mathbf{x};\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{h}\in\mathfrak{G}}\sum\limits_{\begin{subarray}{c}O\in
A(G)\\\
\mathfrak{g}\in\mathfrak{G}_{O}\end{subarray}}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(\mathcal{A}(G))}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{g}).$

The first equality is a formula for computing induced characters. The second
equality comes from the Orbit-Stabilizer Theorem, and changing the order of
summation. The third equality is due to the fact that our quasisymmetric
functions are invariants, so
$\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{hg}\mathfrak{h}^{-1})=\Omega(P_{\mathfrak{h}^{-1}O},\mathfrak{G}_{\mathfrak{h}^{-1}O},\mathbf{x};\mathfrak{g}).$
Finally, we can reindex our summation by replacing $O$ with $\mathfrak{h}O$.
Since $A(G)$ is invariant under $\mathfrak{G}$, we end up with the same number
of terms in the sum. Since
$\operatorname{asc}(O)=\operatorname{asc}(\mathfrak{g}O)$, we obtain the
fourth equality. The last equality is then immediate. ∎

##  5\. Combinatorial Reciprocity Theorem

As stated in the introduction, to us the setting of a combinatorial
reciprocity theorem consists of a subspace $V$ of a ring of formal power
series in some number of variables, and an involution $\omega$ on $V$. Then
given two generating functions $f$ and $g$, a combinatorial reciprocity
theorem is the claim that $f=\omega g$.

For example, let $V$ be the vector space of polynomial functions of degree
$d$. Given a polynomial $p(x)$, we can consider the formal power series
$\sum\limits_{n\geq 0}p(n)t^{n}$. In this way, $V$ is a subspace of the ring
of formal power series in $t$. We also know that $V$ is part of a Hopf
algebra, and so $\omega=(-1)^{d}S$ is an involution on $V$. This is the
setting for many combinatorial reciprocity theorems in the literature.

Similarly, one can let $V$ be the vector space of quasisymmetric functions (or
symmetric functions) of degree $d$, and let $\omega=(-1)^{d}S$, where $S$ is
the antipode. There are several examples of combinatorial reciprocity theorems
involving $\operatorname{QSYM}$ or sym, including Theorem 4.2 of Stanley [18]
or Theorem 4.2 of Grinberg [10].

###  5.1. Combinatorial Reciprocity for Double Posets

Given a double poset $D$, the dual poset $D^{\ast}$ is obtained by reversing
the first partial order $\leq_{1}$ of $D$. We prove a combinatorial
reciprocity theorem for locally special double posets. The orbital version of
the combinatorial reciprocity theorem was previously obtained by Grinberg
[10]. We also prove a combinatorial reciprocity theorem for the chromatic
quasisymmetric class function of a digraph.

######  Theorem 11.

Let $D$ be a locally special double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$.

Then
$(-1)^{|N|}\operatorname{sgn}S\Omega(D,\mathfrak{G},\mathbf{x})=\Omega(D^{\ast},\mathfrak{G},\mathbf{x})$.

Also,
$(-1)^{|N|}\operatorname{sgn}\Omega(D,\mathfrak{G},-x)=\Omega(D^{\ast},\mathfrak{G},x)$.

The second result follows from the first via principal specialization. Our
proof relies a lot on the work of Grinberg [10].

First, we discuss a weighted generalization of
$\Omega(D,\mathfrak{G},\mathbf{x})$. Given a weight function
$\mathbf{w}:N\to\mathbb{N}$, a double poset $D$ on $N$, and a $D$-partition
$\sigma$, define

$\mathbf{x}^{\mathbf{w},\sigma}=\prod_{i\in N}x_{\sigma(i)}^{\mathbf{w}(i)}.$

We let
$\Omega(D,\mathbf{w},\mathbf{x})=\sum\limits_{\sigma}\mathbf{x}^{\mathbf{w},\sigma}$
where the sum is over all $D$-partitions. The following is Theorem 4.2 of
Grinberg [10]:

######  Proposition 12.

Let $D$ be a double poset on a finite set $N$, and let
$\mathbf{w}:N\to\mathbb{N}$. If $D$ is locally special, then

$(-1)^{|N|}S\Omega(D,\mathbf{w},\mathbf{x})=\Omega(D^{\ast},\mathbf{w},\mathbf{x}).$

We let $\operatorname{Aut}(D,\mathbf{w})$ be the set of automorphisms
$\mathfrak{g}$ of $D$ with the property that
$\mathbf{w}\circ\mathfrak{g}=\mathbf{w}$. Let $\mathfrak{G}$ be a subgroup of
$\operatorname{Aut}(D,\mathbf{w})$. Then $\mathfrak{G}$ acts on the set of
$D$-partitions. We let
$\Omega(D,w,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum\limits_{\sigma}\mathbf{x}^{\mathbf{w},\sigma}$,
where the sum is over all $D$-partitions $\sigma$ with the property that
$\sigma\circ\mathfrak{g}=\sigma$. Since
$\mathfrak{g}\in\operatorname{Aut}(G,\mathbf{w})$, the resulting power series
is a quasisymmetric function.

Given $\mathfrak{g}\in\mathfrak{G}$, let $\operatorname{Cyc}(\mathfrak{g})$ be
the set of cycles of $\mathfrak{g}$. We can define a new weight function
$\mathbf{w}/\mathfrak{g}:\operatorname{Cyc}(\mathfrak{g})\to\mathbb{N}$ by
$\mathbf{w}/\mathfrak{g}(C)=\sum\limits_{x\in C}\mathbf{w}(x)$. Given two
cycles $C_{1}$ and $C_{2}$ of $\operatorname{Cyc}(\mathfrak{g})$, and
$i\in\\{1,2\\}$, we say $C_{1}\leq_{i}C_{2}$ if there exists $x\in C_{1}$ and
$y\in C_{2}$ such that $x\leq_{i}y$. This turns
$\operatorname{Cyc}(\mathfrak{g})$ into a double poset that we denote by
$D/\mathfrak{g}$. The following is a combination of parts of Propositions 7.5
and 7.6 of Grinberg [10].

######  Lemma 13.

Let $D$ be a double poset on a finite set $N$, let
$\mathbf{w}:N\to\mathbb{N}$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D,\mathbf{w})$. Given
$\mathfrak{g}\in\mathfrak{G}$, we have

$\Omega(D,w,\mathfrak{G},\mathbf{x};\mathfrak{g})=\Omega(D/\mathfrak{g},w/\mathfrak{g},\mathfrak{G},\mathbf{x}).$

Moreover, if $D$ is locally special, then so is $D/\mathfrak{g}$.

Finally, we let $\mathbf{1}:N\to\mathbb{N}$ be the function defined by
$\mathbf{1}(n)=1$ for all $n\in N$.

###### Proof of Theorem 11.

We have

$\displaystyle(-1)^{|N|}S\operatorname{sgn}(\mathfrak{g})\Omega(D,1,\mathfrak{G},\mathbf{x};\mathfrak{g})$
$\displaystyle=(-1)^{|N|}(-1)^{|N|-\operatorname{cyc}(\mathfrak{g})}S\Omega(D/\mathfrak{g},1/\mathfrak{g},\mathfrak{G},\mathbf{x})$
$\displaystyle=\Omega((D/\mathfrak{g})^{\ast},1/\mathfrak{g},\mathfrak{G},\mathbf{x})$
$\displaystyle=\Omega(D^{\ast}/\mathfrak{g},1/\mathfrak{g},\mathfrak{G},\mathbf{x})$
$\displaystyle=\Omega(D^{\ast},1,\mathfrak{G},\mathbf{x};\mathfrak{g})$

where the first and last equalities are due to Lemma 13, the second equality
is due to Proposition 12. ∎

###  5.2. Combinatorial Reciprocity Theorem for Digraphs

Now we discuss a combinatorial reciprocity theorem for directed graphs. Given
a digraph $G$ on $N$, an acyclic coloring is a pair $(O,f)$ satisfying:

  1. (1)

$O$ is an acyclic orientation of $G$

  2. (2)

For every edge $(u,v)\in E(O)$, $f(u)\leq f(v)$.

We modify the definition of descent. An $O$-descent is an edge $(u,v)\in E(G)$
where $(v,u)\in E(O)$. Let $Y_{G}$ be the acyclic colorings. Then
$\mathfrak{G}$ acts on $Y_{G}$. For $\mathfrak{g}\in\mathfrak{G}$, define:

$\overline{\chi}(G,\mathfrak{G},t,\mathbf{x})=\sum\limits_{(O,f)\in\operatorname{Fix}_{\mathfrak{g}}(Y_{G})}t^{\operatorname{des}(f)}\mathbf{x}^{f}.$

Then $\overline{\chi}(G,\mathfrak{G},t,\mathbf{x})$ is a class function, and
hence is a quasisymmetric class function.

######  Theorem 14.

Let $G$ be a digraph and let $\mathfrak{G}\subseteq\operatorname{Aut}(G)$.
Then

$(-1)^{|N|}S\operatorname{sgn}\chi(G,\mathfrak{G},t,\mathbf{x})=\overline{\chi}(G,\mathfrak{G},t,\mathbf{x})$

and

$(-1)^{|N|}\operatorname{sgn}\chi(G,\mathfrak{G},t,-x)=\overline{\chi}(G,\mathfrak{G},t,x).$

###### Proof.

Let $\mathfrak{g}\in\mathfrak{G}$. Then

$\displaystyle(-1)^{|N|}S\operatorname{sgn}\chi(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(\mathcal{A}(G))}t^{\operatorname{asc}(O)}(-1)^{|N|}S\operatorname{sgn}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(\mathcal{A}(G))}t^{\operatorname{asc}(O)}\Omega(P_{O}^{\ast},\mathfrak{G}_{O},\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum\limits_{O\in\operatorname{Fix}_{\mathfrak{g}}(\mathcal{A}(G))}t^{\operatorname{asc}(O)}\sum\limits_{f:(\overleftarrow{O},f)\in\operatorname{Fix}_{\mathfrak{g}}(Y_{G})}\mathbf{x}^{f}$
$\displaystyle=\sum\limits_{(O,f)\in\operatorname{Fix}_{\mathfrak{g}}(Y_{G})}t^{\operatorname{des}(O)}\mathbf{x}^{f}=\overline{\chi}(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g}).$

where the first equality follows from Lemma 10, and the second equality is due
to Theorem 11. The third equality comes from observing that $f\in
P_{P_{O}^{\ast}}$ if and only if $(\overleftarrow{O},f)$ is an acyclic
coloring. The fourth equality comes from observing that
$\operatorname{asc}(O)=\operatorname{des}(\overleftarrow{O})$, and reindexing
the summation by substituting $O$ with $\overleftarrow{O}$. ∎

##  6\. Flawlessness

In this section, we study the property of being $M$-increasing or effectively
flawless. Let $D$ be a double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. For $\alpha\models|N|$, let
$V_{\alpha,D}$ be the vector space with basis $X_{\alpha,D}$. Then
$V_{\alpha,D}$ is a $\mathfrak{G}$-module.

Given $\alpha\leq\beta\models|N|$, we define $\mathfrak{G}$-invariant
injective maps $\theta_{\alpha,\beta}:V_{\alpha,D}\to V_{\beta,D}$. Given a
$D$-set composition $C$ of type $\alpha$, we let

(2) $\theta_{\alpha,\beta}(C)=\sum\limits_{C^{\prime}\in
X_{\beta,D}:C^{\prime}\geq C}C^{\prime}.$

######  Proposition 15.

Let $D$ be a double poset on a finite set $N$. Let $\alpha\leq\beta\leq\gamma$
be integer compositions of $|N|$. Then $\theta_{\alpha,\beta}$ is an injective
$\mathfrak{G}$-invariant map. Moreover, we have
$\theta_{\gamma,\beta}\circ\theta_{\alpha,\beta}=\theta_{\gamma,\alpha}$.

###### Proof.

We see that $\theta_{\alpha,\beta}$ is $\mathfrak{G}$-invariant. To see that
the map is injective, it is enough to note that the map $f:V_{\beta,D}\to
V_{\alpha,D}$ given by $f(C)=C_{\alpha}(C)$ is a section for
$\theta_{\alpha,\beta}$. Hence $\theta_{\alpha,\beta}$ is injective and $f$ is
surjective.

Let $\alpha\leq\beta\leq\gamma$, and let $C\in X_{\alpha,D}$. Then

$\displaystyle\theta_{\beta,\gamma}\circ\theta_{\alpha,\beta}(C)$
$\displaystyle=\sum\limits_{C^{\prime}\in X_{\beta,D}:C^{\prime}\geq
C}\theta_{\beta,\gamma}(C^{\prime})$ $\displaystyle=\sum\limits_{C^{\prime}\in
X_{\beta,D}:C^{\prime}\geq C}\sum\limits_{C^{\prime\prime}\in
X_{\gamma,D}:C^{\prime\prime}\geq C^{\prime}}C^{\prime\prime}$
$\displaystyle=\sum\limits_{C^{\prime\prime}\in
X_{\gamma,D}:C^{\prime\prime}\geq C}\sum\limits_{C^{\prime}\in
X_{\beta,D}:C^{\prime\prime}\geq C^{\prime}\geq C}C^{\prime\prime}$
$\displaystyle=\sum\limits_{C^{\prime\prime}\in
X_{\gamma,D}:C^{\prime\prime}\geq
C}C^{\prime\prime}=\theta_{\alpha,\gamma}(C).$

The penultimate equality follows from the fact that there is exactly one set
composition $C^{\prime}$ of type $\beta$ with the property that $C\leq
C^{\prime}\leq C^{\prime\prime}$. Moreover, if $C$ and $C^{\prime\prime}$ are
$D$-set compositions, then $C^{\prime}$ is as well. Hence the inner summation
only involves one term. ∎

######  Theorem 16.

Let $D$ be a double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Then
$\Omega(D,\mathfrak{G},\mathbf{x})$ is $M$-increasing and
$\Omega(D,\mathfrak{G},x)$ is effectively flawless.

Let $G$ be a digraph, and let $\mathfrak{H}\subseteq\operatorname{Aut}(G)$.
For $k\in\mathbb{N}$, we have $[t^{k}]\chi(G,\mathfrak{H},t,\mathbf{x})$ is
$M$-increasing and $[t^{k}]\chi(G,\mathfrak{H},t,x)$ is effectively flawless.

###### Proof.

Let $D$ be a double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. For $\alpha\models|N|$, we see
that $[M_{\alpha}]\Omega(D,\mathfrak{G},\mathbf{x})$ is the character of the
representation of $\mathfrak{G}$ on $V_{\alpha,D}$. Let
$\alpha\leq\beta\models|N|$. Since $\theta_{\alpha,\beta}$ is injective and
$\mathfrak{G}$-invariant, $V_{\alpha,D}$ is isomorphic to a submodule of
$V_{\beta,D}$. Thus
$[M_{\beta}]\Omega(D,\mathfrak{G},\mathbf{x})-[M_{\alpha}]\Omega(D,\mathfrak{G},\mathbf{x})$
is the character corresponding to the complement of
$\theta_{\alpha,\beta}(V_{\alpha,D})$ in $V_{\beta,D}$. Hence
$\Omega(D,\mathfrak{G},\mathbf{x})$ is $M$-increasing.

Since $\Omega(D,\mathfrak{G},\mathbf{x})$ is $M$-increasing and
$\Omega(D,\mathfrak{G},x)$ is the principal specialization of
$\Omega(D,\mathfrak{G},\mathbf{x})$, it follows from Proposition 3 (4) that
$\Omega(D,\mathfrak{G},x)$ is effectively flawless.

Let $G$ be a directed graph, and let
$\mathfrak{H}\subseteq\operatorname{Aut}(G)$. By Theorem 19, we see that
$[t^{k}]\chi(G,\mathfrak{H},t,\mathbf{x})$ is $F$-effective. Thus
$[t^{k}]\chi(G,\mathfrak{H},t,\mathbf{x})$ is $M$-increasing. Since
$\operatorname{ps}(\chi(G,\mathfrak{H},t,\mathbf{x}))=\chi(G,\mathfrak{H},t,x)$,
it follows that $[t^{k}]\chi(G,\mathfrak{G},t,x)$ is effectively flawless. ∎

$b$$d$$a$$c$$e$ Figure 4. A Hasse diagram.

Now we discuss counterexamples to extending Theorem 16 to studying
coefficients in the $F$ basis and to studying the $h$-vector. Let $D$ be the
poset in Figure 4. We define $\leq_{2}$ to be the opposite partial order of
$\leq_{1}$. Hence $\Omega(D,x)$ counts strict $D$-partitions. We let
$\mathfrak{G}=\mathbb{Z}_{2}\times\mathbb{Z}_{3}$, viewed as the group
$\langle(abc),(de)\rangle$. Let $\omega$ be a cubed root of unity. If we let
$\sigma=(abc)$ and $\tau=(de)$, then Table 1 is the character table for
$\mathfrak{G}$:

$\begin{array}[]{c|cccccc}&\\{e\\}&\\{\sigma\\}&\\{\sigma^{2}\\}&\\{\tau\\}&\\{\tau\sigma\\}&\\{\tau\sigma^{2}\\}\\\
\hline\cr\chi_{1}&1&1&1&1&1&1\\\
\chi_{2}&1&\omega&\bar{\omega}&1&\omega&\bar{\omega}\\\
\chi_{3}&1&\bar{\omega}&\omega&1&\bar{\omega}&\omega\\\
\chi_{4}&1&1&1&-1&-1&-1\\\
\chi_{5}&1&\omega&\bar{\omega}&-1&-\omega&-\bar{\omega}\\\
\chi_{6}&1&\bar{\omega}&\omega&-1&-\bar{\omega}&-\omega\end{array}$

Table 1. The character table for $\mathbb{Z}_{3}\times\mathbb{Z}_{2}$

Then

$\displaystyle\Omega(D,\mathfrak{G},\mathbf{x})$
$\displaystyle=\chi_{1}F_{3,2}+(\chi_{2}+\chi_{3})(F_{1,2,2}+F_{2,1,2})+\chi_{4}F_{3,1,1}$
$\displaystyle+(\chi_{5}+\chi_{6})(F_{1,2,1,1}+F_{2,1,1,1})+\chi_{1}F_{1,1,1,2}+\chi_{4}F_{1,1,1,1,1}.$

Thus $\Omega(D,\mathfrak{G},\mathbf{x})$ is _not_ $F$-increasing. We see that

$\sum_{m\geq
0}\Omega(D,\mathfrak{G},m)t^{m}=\frac{\chi_{1}t^{2}+(2\chi_{2}+2\chi_{3}+\chi_{4})t^{3}+(2\chi_{5}+2\chi_{6}+\chi_{1})t^{4}+\chi_{4}t^{5}}{(1-t)^{6}}.$

Then the $h$-vector is
$(0,0,\chi_{1},2\chi_{2}+2\chi_{3}+\chi_{4},2\chi_{5}+2\chi_{6}+\chi_{1},\chi_{4})$.
We see that $h_{2}$ and $h_{3}$ are not comparable, so the $h$-vector is not
strongly flawless.

Similarly, we can view the Hasse diagram in Figure 4 as a digraph $G$ with
$\mathfrak{G}=\mathbb{Z}_{3}\times\mathbb{Z}_{2}$. Then
$[t^{0}]\chi(G,\mathfrak{G},\mathbf{x})=\Omega(D,\mathfrak{G},\mathbf{x})$.
Thus we have an example of a directed graph where the chromatic quasisymmetric
class function is not $F$-increasing, and where the $h$-vector of the
chromatic polynomial class function is not effectively flawless.

##  7\. F-effectiveness and $h$-effectiveness

In this section, we state and prove several effectiveness theorems.

######  Theorem 17.

Let $D$ be a locally special double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Then
$\Omega(D,\mathfrak{G},\mathbf{x})$ is $F$-effective.

We prove Theorem 17 in the next subsection. First, we focus on corollaries and
other related theorems. First, we obtain the follow result from Proposition 2
(3).

######  Corollary 18.

Let $D$ be a locally special double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Given an irreducible character
$\psi$, we have $\langle\psi,\Omega(D,\mathfrak{G},\mathbf{x})\rangle$ is
$F$-positive.

We also obtain the following theorem for the chromatic quasisymmetric class
function of a digraph.

######  Theorem 19.

Let $G$ be a digraph, and let $\mathfrak{G}\subseteq\operatorname{Aut}(G)$.
For any $k\in\mathbb{N}$, we have $[t^{k}]\chi(G,\mathfrak{G},\mathbf{x})$ is
$F$-effective.

For any irreducible character $\psi$ of $\mathfrak{G}$, we have
$\langle\psi,[t^{k}]\chi(G,\mathfrak{G},\mathbf{x})\rangle$ is $F$-positive.

The second statement follows from applying Proposition 2 (3) to the first
statement, so we focus on proving the first statement.

###### Proof.

Let $\mathcal{T}$ be a transversal for the action of $\mathfrak{G}$ on
$\mathcal{A}(G)$. For each acyclic orientation $O\in\mathcal{T}$, we write
$\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x})=\sum\limits_{\alpha\models|N|}\psi_{\alpha,P_{O}}F_{\alpha}$
where the characters $\psi_{\alpha,P_{O}}$ are effective. Using Lemma 10, we
have

$\displaystyle\chi(G,\mathfrak{G},t,\mathbf{x})$
$\displaystyle=\sum\limits_{O\in\mathcal{T}}t^{\operatorname{asc}(O)}\Omega(P_{O},\mathfrak{G}_{O},\mathbf{x})\uparrow_{\mathfrak{G}_{O}}^{\mathfrak{G}}$
$\displaystyle\sum\limits_{O\in\mathcal{T}}t^{\operatorname{asc}(O)}\left(\sum\limits_{\alpha\models|N|}\psi_{\alpha,P_{O}}F_{\alpha}\right)\uparrow_{\mathfrak{G}_{O}}^{\mathfrak{G}}$
$\displaystyle=\sum\limits_{\alpha\models|N|}\left(\sum\limits_{O\in\mathcal{T}}t^{\operatorname{asc}(O)}\psi_{\alpha,P_{O}}\uparrow_{\mathfrak{G}_{O}}^{\mathfrak{G}}\right)F_{\alpha}.$

∎

We also deduce results about $h$-effectiveness by applying Proposition 3 (2).

######  Corollary 20.

Let $D$ be a locally special double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D).$ Then $\Omega(D,\mathfrak{G},x)$
is $h$-effective.

Let $G$ be a directed graph. Then $\chi(G,\mathfrak{G},t,x)$ is $h$-effective.

###  7.1. Proof of Theorem 17

For $\alpha\leq\beta\models|N|$, define $\theta_{\alpha,\beta}$ by Equation
(2). Our first step is to prove the following proposition.

######  Proposition 21.

Let $D$ be a double poset on a finite set $N$. Let $\alpha\leq\beta\models|N|$
and $\gamma\leq\beta$. Then we have
$\theta_{\alpha,\beta}(V_{\alpha,D})\cap\theta_{\gamma,\beta}(V_{\gamma,D})=\theta_{\alpha\wedge\gamma,\beta}(V_{\alpha\wedge\gamma,D})$.

Fix a $D$-compatible linear order $\ell$, which exists by Lemma 7. Let
$\alpha\leq\beta$ and $\gamma\leq\beta$. Fix
$\vec{v}\in\theta_{\alpha,\beta}(V_{\alpha,D})\cap\theta_{\gamma,\beta}(V_{\gamma,D}).$
Consider $C\in X_{\beta,D}\cap\theta_{\alpha,\beta}(V_{\alpha,D})$. Then
$C_{\alpha}(C)\in X_{\alpha,D}$. Let $C^{1}=C_{\beta}(\ell(C_{\alpha}(C))).$
We see that $C_{\alpha}(C)\leq C^{1}$, so if $C^{1}$ contained any inversions,
then so does $C_{\alpha}(C)$. Since $C^{1}\leq\ell(C_{\alpha}(C))$, we know
that the union of the first several blocks of $C^{1}$ is the union of the
first several blocks of $\ell(C_{\alpha}(C)).$ Thus $C^{1}$ is a $D$-set
composition.

Since $C_{\alpha}(C)\leq C$ and $C^{1}\leq\ell(C_{\alpha}(C))$, we have
$\ell(C_{1})\leq_{\ell}\ell(C_{\alpha}(C))<_{\ell}\ell(C)$. Thus
$C^{1}\leq_{\ell}C$. Moreover, since
$\vec{v}\in\theta_{\alpha,\beta}(V_{\alpha,D})$, we know
$\vec{v}=\theta_{\alpha,\beta}(\vec{w})$ for some $\vec{w}\in V_{\alpha,D}$.
Since $C_{\alpha}(C)\leq C^{1}$, it follows from the definition of
$\theta_{\alpha,\beta}$ that $[C^{1}]\vec{v}=[C_{\alpha}(C)]\vec{w}$.
Similarly, since $C_{\alpha}(C)\leq C$, we have
$[C]\vec{v}=[C_{\alpha}(C)]\vec{w}$. Thus $[C^{1}]\vec{v}=[C]\vec{v}$.

Let $C^{2}=C_{\beta}(\ell(C_{\gamma}(C^{1}))).$ By similar reasoning,
$C^{2}\leq_{\ell}C^{1}$ and $[C^{2}]\vec{v}=[C^{1}]\vec{v}=[C]\vec{v}$. For
every integer $k$, we define

$C^{k+1}=\begin{cases}C_{\beta}(\ell(C_{\alpha}(C^{k})))&k\mbox{ is even}\\\
C_{\beta}(\ell(C_{\gamma}(C^{k})))&k\mbox{ is odd}\end{cases}$

Thus we obtain a sequence of compositions $C^{k}$ such that
$C^{k+1}\leq_{\ell}C^{k}$ and $[C^{k}]\vec{v}=[C]\vec{v}$ for all $k$.

Let $m$ be the first integer where $C^{m+1}=C^{m}$. Define
$f(C)=C_{\alpha\wedge\gamma}(\ell(C^{m}))$.

######  Lemma 22.

Let
$\vec{v}\in\theta_{\alpha,\beta}(V_{\alpha,D})\cap\theta_{\gamma,\beta}(V_{\gamma,D}).$
Consider $C\in X_{\beta,D}$. Fix $f(C)$ as defined above. Then $f(C)\in
X_{\alpha\wedge\gamma,D}$.

Moreover, given $D$ such that $D\geq f(C)$, we have $f(D)=f(C).$

Before we give the proof, we define another operation on set compositions.
Given $C\models N$, and $S\subset N$, the set composition $C\cap S$ is defined
by taking $(C_{1}\cap S,C_{2}\cap S,\cdots,C_{\ell(C)}\cap S)$, and then
deleting any coordinates corresponding to empty intersections. For instance,
$123|456|789\cap\\{1,7,8\\}=1|78$.

###### Proof.

We show that $f(C)$ is a $D$-set composition. Write
$f(C)=C^{\prime}_{1}|\cdots|C^{\prime}_{r}$. Since $f(C)\leq C^{m}$, the set
$C^{\prime}_{1}\cup\cdots\cup C^{\prime}_{i}$ is a union of blocks of $C^{m}$,
and hence is a $\leq_{1}$-order ideal. Thus, it suffices to show that that
$C^{\prime}_{i}$ does not contain any inversions. Define
$D^{\prime}=C_{\alpha}(C^{m})\cap C^{\prime}_{i}$, which is a set composition
of $C^{\prime}_{i}$. Let $\ell^{\prime}$ be the restriction of $\ell(C^{m})$
to $C^{\prime}_{i}$. Then $\ell^{\prime}$ is lexicographically least in
$C^{\prime}_{i}$; if there exists $\tau<_{\ell}\ell^{\prime}$, then we could
modify $\ell(C^{m})$ by replacing $\ell(C^{m})|_{C^{\prime}_{i}}$ with $\tau$
and obtain a new refinement of $C^{m}$ that is lexicographically smaller than
$\ell(C^{m})$. Since each block $D^{\prime}_{j}$ of $D^{\prime}$ contains no
inversions, by Proposition 8, $\ell^{\prime}|_{D^{\prime}_{j}}$ is strictly
increasing. If we let $E^{\prime}=C_{\gamma}(C^{m})\cap C^{\prime}_{i}$, then
by similar reasoning $\ell^{\prime}$ is strictly increasing when restricted to
each block of $E^{\prime}$. We see that the finest common coarsening of
$D^{\prime}$ and $E^{\prime}$ is the set composition $(C_{i})$. Thus
$\ell(C^{m})$ is strictly increasing on all of $C^{\prime}_{i}$. By
Proposition 8, it follows that $D$ has no inversions in $C^{\prime}_{i}$.
Therefore $C^{\prime}$ is a $D$-set composition. We also see that
$C^{m}=C_{\gamma}(\ell(f(C))).$

Now let $D\geq f(C)$. For every integer $k$, we define

$D^{k+1}=\begin{cases}C_{\beta}(\ell(C_{\alpha}(D^{k})))&k\mbox{ is even}\\\
C_{\beta}(\ell(C_{\gamma}(D^{k})))&k\mbox{ is odd}\end{cases}$

Thus we obtain a sequence of compositions $D^{k}$ such that
$D^{k+1}\leq_{\ell}D^{k}$ and $[D^{k}]\vec{v}=[D]\vec{v}$ for all $k$. We
observe that $D^{k+1}$ involves permuting the elements from blocks of $D^{k}$
that belong to the same block of $C_{\alpha}(D^{k})$ (or $C_{\gamma}(C^{k})$).
Hence $C_{\alpha\wedge\gamma}(D^{k+1})=C_{\alpha\wedge\gamma}(D^{k})=f(C)$ for
all $k$. Thus $D^{k+1}\geq f(C)$ for all $k$.

If we let $m$ be the first integer for which $D^{m+1}=D^{m}$, then we have
$f(D)=C_{\alpha\wedge\gamma}(D^{m})=f(C)$. ∎

###### Proof of Proposition 21.

Let
$\vec{v}\in\theta_{\alpha,\beta}(V_{\alpha,D})\cap\theta_{\gamma,\beta}(V_{\gamma,D}).$
For any $C\in X_{\beta,D}$, let $m(C)=C_{\beta}(\ell(f(C)))$. Then we have
$\vec{v}_{C}=\vec{v}_{m(C)}$. Moreover, for any $C^{\prime}\geq f(C)$, by
Lemma 22, we have $f(C)=f(C^{\prime})$. Hence $m(C^{\prime})=m(C)$ and
$[C]\vec{v}=[C^{\prime}]\vec{v}$. Given $D\in X_{\alpha\wedge\gamma,D}$, we
define $[D]\vec{w}=[C]\vec{v}$ for any $C\in X_{\beta,D}$ such that $f(C)=D$.
By what we have just observed, $[D]\vec{w}$ does not depend upon the choice of
$C$. Also, for any $C\in X_{\beta,D}$, we know that $f(C)\in
X_{\alpha\wedge\gamma,D}$ by Lemma 22. Thus
$\theta_{\alpha\wedge\gamma,\beta}(\vec{w})=\vec{v}$. Hence
$\vec{v}\in\theta_{\alpha\wedge\gamma,\beta}(V_{\alpha\wedge\gamma,D})$. ∎

For $\alpha\models|N|$, we define $W_{\alpha,D}$ to be the complement in
$V_{\alpha,D}$ of the module

$\operatorname{span}\\{\theta_{\beta,\alpha}(X_{\beta,D}):\beta<\alpha\\}$

######  Proposition 23.

Let $D$ be a locally special double poset on a finite set $N$. Let
$\alpha\leq\beta\models|N|$ and $\gamma\leq\beta$. Then we have
$\theta_{\alpha,\beta}(W_{\alpha,D})\cap\theta_{\gamma,\beta}(W_{\gamma,D})=\\{\vec{0}\\}$.

###### Proof.

Let
$\vec{v}\in\theta_{\alpha,\beta}(W_{\alpha,D})\cap\theta_{\gamma,\beta}(W_{\gamma,D}).$
Then $\vec{v}=\theta_{\alpha,\beta}(\vec{y})$ for some $\vec{y}\in
W_{\alpha,D}$. By Proposition 21 and Proposition 15, we have
$\vec{v}\in\theta_{\alpha\wedge\gamma,\beta}(V_{\alpha\wedge\gamma,D})=\theta_{\alpha,\beta}\circ\theta_{\alpha\wedge\gamma,\alpha}(V_{\alpha\wedge\gamma,D}).$
Thus $\vec{v}=\theta_{\alpha,\beta}(\vec{z})$ for some
$\vec{z}\in\theta_{\alpha\wedge\gamma,\alpha}(V_{\alpha\wedge\gamma,D}).$
Since $\theta_{\alpha,\beta}$ is injective, we have $\vec{y}=\vec{z}$.
However, by the definition of $W_{\alpha,D}$, it follows that
$\vec{y}=\vec{z}=\vec{0}$, and thus $\vec{v}=\vec{0}$. Hence
$\theta_{\alpha,\beta}(W_{\alpha,D})\cap\theta_{\gamma,\beta}(W_{\gamma,D})=\\{\vec{0}\\}$.
∎

###### Proof of Theorem 17.

Let $D$ be a locally special poset on $N$ with
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Let us write
$\Omega(D,\mathfrak{G},\mathbf{x})=\sum\limits_{\alpha\models|N|}\psi_{\alpha,D}F_{\alpha}$
where the $\psi_{\alpha,D}$ are virtual characters. Then we have
$\chi_{\alpha}(D,\mathfrak{G})=\sum\limits_{\beta\leq\alpha}\psi_{\beta,D}$.
We claim that $\psi_{\beta,D}$ is the character of $W_{\beta,D}$. It suffices
to show that

(3)
$V_{\alpha,D}=\bigoplus_{\beta\leq\alpha}\theta_{\beta,\alpha}(W_{\beta,D})\simeq\bigoplus_{\beta\leq\alpha}W_{\beta,D}$

as $\mathfrak{G}$-modules for all $\alpha\models|N|$.

We prove Equation (3) by induction on $\ell(\alpha)$. We see that

$\operatorname{span}\\{\theta_{\beta,\alpha}(W_{\beta,D}):\beta<\alpha\\}\subseteq\operatorname{span}\\{\theta_{\beta,\alpha}(X_{\beta,D}):\beta<\alpha\\}.$

By induction, we have

$\theta_{\beta,\alpha}(V_{\beta,D})=\theta_{\beta,\alpha}(\bigoplus_{\gamma\leq\beta}\theta_{\gamma,\beta}(W_{\gamma,D}))=\bigoplus_{\gamma\leq\beta}\theta_{\gamma,\alpha}(W_{\gamma,D}).$

The last equality uses the fact that $\theta_{\beta,\alpha}$ is
$\mathfrak{G}$-invariant, and the property that
$\theta_{\beta,\alpha}\circ\theta_{\gamma,\beta}=\theta_{\gamma,\alpha}$. Thus
we have

$\operatorname{span}\\{\theta_{\beta,\alpha}(X_{\beta,D}):\beta<\alpha\\}=\operatorname{span}\\{\theta_{\beta,\alpha}(W_{\beta,D}):\beta<\alpha\\}=\bigoplus_{\beta<\alpha}\theta_{\beta,\alpha}(W_{\beta,D}).$

Combined with the definition of $W_{\alpha,D}$, we get

$V_{\alpha,D}=\bigoplus_{\beta\leq\alpha}\theta_{\beta,\alpha}(W_{\beta,D})\simeq\bigoplus_{\beta\leq\alpha}W_{\beta,D}$

where the last isomorphism comes from the fact that $\theta_{\beta,\alpha}$ is
$\mathfrak{G}$-invariant and injective. ∎

##  8\. Orbital Invariants

In this section we define orbital quasisymmetric function invariants. In the
case of double posets, the resulting invariant was already studied by Grinberg
[10]. These are quasisymmetric functions whose coefficients count the number
of orbits of a group action. Due to Burnside’s Lemma, these invariants can be
computed as $\langle 1,\chi\rangle$, where $1$ is the trivial character, and
$\chi$ is the quasisymmetric class function. As a result, we see that we
derive many results for our orbital invariants from the class functions.

We define the _orbital quasisymmetric $D$-partition enumerator_ by

$\Omega^{O}(D,\mathfrak{G},\mathbf{x})=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})$

and the _orbital order polynomial_ by

$\Omega^{O}(D,\mathfrak{G},x)=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\Omega(D,\mathfrak{G},x;\mathfrak{g}).$

We also define the _orbital chromatic quasisymmetric function_ of a digraph
$G$ by

$\chi^{O}(G,\mathfrak{G},t,\mathbf{x})=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\chi(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g})$

and the _orbital chromatic polynomial_ by

$\chi^{O}(G,\mathfrak{G},t,x)=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\chi(G,\mathfrak{G},t,x;\mathfrak{g})$

The following results follow from Burnside’s Lemma. Property 2 is Grinberg’s
definition of the orbital $D$-partition enumerator.

######  Proposition 24.

Let $D$ be a double poset on a finite set $N$ and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Let $G$ be a digraph on $N$, and
let $\mathfrak{H}\subseteq\operatorname{Aut}(G)$.

  1. (1)

We have
$[M_{\alpha}]\Omega^{O}(D,\mathfrak{G},x)=|X_{\alpha,D}/\mathfrak{G}|$.

  2. (2)

Let $\mathcal{T}$ be a transversal for $\mathfrak{G}$ acting on $P_{D}$. Then
$\Omega^{O}(D,\mathfrak{G},\mathbf{x})=\sum\limits_{f\in\mathcal{T}}\mathbf{x}^{f}$.

  3. (3)

For $n\in\mathbb{N}$, we have
$\Omega^{O}(D,\mathfrak{G},n)=|X_{n,D}/\mathfrak{G}|$.

  4. (4)

Let $\mathcal{T}$ be a transversal for $\mathfrak{H}$ acting on $C_{G}$. Then
$\Omega^{O}(G,\mathfrak{H},\mathbf{x})=\sum\limits_{f\in\mathcal{T}}\mathbf{x}^{f}$.

  5. (5)

For $n\in\mathbb{N}$, and $k>0$, we have
$[t^{k}]\chi^{O}(G,\mathfrak{H},t,n)=|C_{k,n,G}/\mathfrak{H}|$.

###### Proof.

We will prove the first and second claim. The rest are similar. For the first,
we observe that

$[M_{\alpha}]\Omega^{O}(D,\mathfrak{G},\mathbf{x})=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}[M_{\alpha}]\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}|\operatorname{Fix}_{\mathfrak{g}}(X_{\alpha,D})|.$

For the second identity, we note that for any $f:N\to\mathbb{N}$ and
$\sigma\in\mathfrak{S}_{N}$, we have $\mathbf{x}^{\sigma f}=\mathbf{x}^{f}$.
We see that

$\displaystyle\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\sum\limits_{f\in\operatorname{Fix}_{\mathfrak{g}}(X_{D})}\mathbf{x}^{f}$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{f\in
X_{D}}\mathbf{x}^{f}\sum\limits_{\mathfrak{g}\in\mathfrak{G}_{f}}1$
$\displaystyle=\sum\limits_{f\in
X_{D}}\frac{1}{|\mathfrak{G}(f)|}\mathbf{x}^{f}$
$\displaystyle=\sum\limits_{h\in\mathcal{T}}\mathbf{x}^{h}\sum\limits_{f\in\mathfrak{G}(h)}\frac{1}{|\mathfrak{G}_{f}|}.$

The second equality is just a rearrangment of terms. The third equality
follows from the Orbit-Stabilizer Theorem. The last equality involves
splitting the summation into a double sum, and recognize that $\mathbf{x}^{f}$
is constant on orbits. Finally, the inner summation simplifies to 1. ∎

We also obtain several facts about the coefficients of the orbital invariants
with respect to various bases. For polynomials $f(t)$ and $g(t)$, we write
$f(t)\leq_{t}g(t)$ if $g(t)-f(t)\in\mathbb{N}[t]$.

######  Corollary 25.

Let $D$ be a double poset on $N$. Given
$\mathfrak{G}\subseteq\operatorname{Aut}(D),$ we have
$\Omega^{O}(D,\mathfrak{G},\mathbf{x})$ is $M$-increasing, and
$\Omega^{O}(D,\mathfrak{G},x)$ is strongly flawless.

###### Proof.

Let $D$ be a double poset on $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Since
$\Omega(D,\mathfrak{G},\mathbf{x})$ is $M$-increasing, then if we take the
inner product with the trivial character, Proposition 2 (4) implies that that
$\Omega^{O}(D,\mathfrak{G},\mathbf{x})$ is $M$-increasing. Also, Proposition 3
(5) implies that $\Omega^{O}(D,\mathfrak{G},x)$ is strongly flawless.

Suppose that $D$ is locally special. Then by Corollary 18, applied with $\psi$
being the trivial character, we see that
$\Omega^{O}(D,\mathfrak{G},\mathbf{x})$ is $F$-positive. Since
$\Omega^{O}(D,\mathfrak{G},x)=\operatorname{ps}(\Omega^{O}(D,\mathfrak{G},\mathbf{x}))$,
it follows from Proposition 3 (5) that $\Omega^{O}(D,\mathfrak{G},x)$ is
$h$-positive.

Let $G$ be a directed graph, and let
$\mathfrak{H}\subseteq\operatorname{Aut}(G)$. Fix $k\in\mathbb{N}$. By Theorem
19, we see that $[t^{k}]\chi^{O}(G,\mathfrak{H},\mathbf{x})$ is $F$-positive.
Hence it is also $M$-increasing. By principal specialization and Proposition 3
(5), we see that $[t^{k}]\chi^{O}(G,\mathfrak{H},x)$ is $h$-positive, and
$[t^{k}]\chi^{O}(G,\mathfrak{H},x)$ is strongly flawless.

∎

###  8.1. Orbital Combinatorial Reciprocity Results

We can also obtain a combinatorial reciprocity for the orbital $D$-partition
enumerator, although it involves the notion of coeven $D$-partition. Given a
group $\mathfrak{G}\subseteq\mathfrak{S}_{N}$, and an action of $\mathfrak{G}$
on a set $X$, let we say that an element $x\in X$ is $\mathfrak{G}$-coeven if
the stabilizer subgroup $\mathfrak{G}_{x}$ is a subgroup of the alternating
group $\mathfrak{A}_{N}$. Let $X^{+}$ be the set of $\mathfrak{G}$-coeven
elements. Then $\mathfrak{G}$ acts on $X^{+}$.

The following Lemma is essentially a restatement of Lemma 7.7 in Grinberg
[10], although our proof is slightly different.

######  Lemma 26.

Let $\mathfrak{G}\subseteq\mathfrak{S}_{N}$ be a group acting on a finite set
$X$.

$|X^{+}/\mathfrak{G}|=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\operatorname{sgn}(\mathfrak{g})|\operatorname{Fix}_{\mathfrak{g}}(X)|$

###### Proof.

We have

$\displaystyle\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\operatorname{sgn}(\mathfrak{g})|\operatorname{Fix}_{\mathfrak{g}}(X)|$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\sum\limits_{x\in
X:\mathfrak{g}x=x}\operatorname{sgn}(\mathfrak{g})$
$\displaystyle=\sum\limits_{x\in
X}\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}_{x}}\operatorname{sgn}(\mathfrak{g}).$

Suppose that $\mathfrak{G}_{x}\not\subseteq\mathfrak{A}_{N}$. Then
$\mathfrak{H}=\mathfrak{A}_{N}\cap\mathfrak{G}$ is a normal subgroup of
$\mathfrak{G}$ of index $2$. Thus half the elements $\mathfrak{g}$ of
$\mathfrak{G}_{x}$ are even, and half are odd, and these elements have
opposite signs under $\operatorname{sgn}$. Hence the inner sum is zero in that
case.

Hence we are left with those $x$ for which
$\mathfrak{G}_{x}\subseteq\mathfrak{A}_{N}$. Then we obtain:

$\displaystyle\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\operatorname{sgn}(\mathfrak{g})|\operatorname{Fix}_{\mathfrak{g}}(X)|$
$\displaystyle=\sum\limits_{x\in
X^{+}}\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}_{x}}\operatorname{sgn}(\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\sum\limits_{x\in\operatorname{Fix}_{\mathfrak{g}}(X^{+})}1$
$\displaystyle=|X^{+}/\mathfrak{G}|.$

∎

We define the _$\mathfrak{G}$ -coeven quasisymmetric function_ by

$\Omega^{+}(D,\mathfrak{G},\mathbf{x})=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\operatorname{sgn}(\mathfrak{g})\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})$

and the _orbital polynomial_ by

$\Omega^{+}(D,\mathfrak{G},x)=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\operatorname{sgn}(\mathfrak{g})\Omega(D,\mathfrak{G},x;\mathfrak{g}).$

By using Lemma 26, we obtain the following results.

######  Proposition 27.

Let $D$ be a double poset on a finite set $N$ and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Let $G$ be a digraph on $N$, and
let $\mathfrak{H}\subseteq\operatorname{Aut}(G)$.

  1. (1)

We have
$[M_{\alpha}]\Omega^{+}(D,\mathfrak{G},x)=|X^{+}_{\alpha,D}/\mathfrak{G}|$.

  2. (2)

Let $\mathcal{T}$ be a transversal for $\mathfrak{G}$ acting on $P_{D}^{+}$.
Then
$\Omega^{+}(D,\mathfrak{G},x)=\sum\limits_{f\in\mathcal{T}}\mathbf{x}^{f}$.

  3. (3)

For $n\in\mathbb{N}$, we have
$\Omega^{+}(D,\mathfrak{G},n)=|X^{+}_{n,D}/\mathfrak{G}|$.

  4. (4)

If $\mathcal{T}$ is a transversal for $\mathfrak{H}$ acting on $C_{D}^{+}$,
then
$\chi^{+}(G,\mathfrak{H},t,n)=\sum\limits_{f\in\mathcal{T}}t^{\operatorname{asc}(f)}\mathbf{x}^{f}$.

  5. (5)

For $n\in\mathbb{N}$, and $k>0$, we have
$[t^{k}]\chi^{+}(G,\mathfrak{H},t,n)=|C^{+}_{k,n,G}/\mathfrak{H}|$.

Now we discuss the Combinatorial Reciprocity Theorem for orbital
quasisymmetric functions. The first result is Theorem 4.7 of Grinberg [10].

######  Theorem 28.

Let $D$ be a locally special double poset on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$. Then we have the following
identities.

  1. (1)

$(-1)^{|N|}S\Omega^{O}(D,\mathfrak{G},\mathbf{x})=\Omega^{+}(D^{\ast},\mathfrak{G},\mathbf{x}).$

  2. (2)

$(-1)^{|N|}S\Omega^{+}(D,\mathfrak{G},\mathbf{x})=\Omega^{O}(D^{\ast},\mathfrak{G},\mathbf{x}).$

  3. (3)

$(-1)^{|N|}\Omega^{O}(D,\mathfrak{G},-x)=\Omega^{+}(D^{\ast},\mathfrak{G},x)$.

  4. (4)

$(-1)^{|N|}\Omega^{+}(D,\mathfrak{G},-x)=\Omega^{O}(D^{\ast},\mathfrak{G},x)$.

Now we discuss the Combinatorial Reciprocity Theorem for orbital chromatic
quasisymmetric functions.

######  Theorem 29.

Let $G$ be a digraph on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(G)$.

Then we have the following identities:

  1. (1)

$(-1)^{|N|}S\chi^{O}(G,\mathfrak{G},t,\mathbf{x})=\overline{\chi}^{+}(G,\mathfrak{G},t,\mathbf{x}).$

  2. (2)

$(-1)^{|N|}S\chi^{+}(G,\mathfrak{G},t,\mathbf{x})=\overline{\chi}^{O}(G,\mathfrak{G},t,\mathbf{x}).$

  3. (3)

$(-1)^{|N|}\chi^{O}(G,\mathfrak{G},t,-x)=\overline{\chi}^{+}(G,\mathfrak{G},t,x)$.

  4. (4)

$(-1)^{|N|}\chi^{+}(G,\mathfrak{G},t,-x)=\overline{\chi}^{O}(G,\mathfrak{G},t,x)$.

###### Proof.

All formulas are proven in a similar manner, so we only prove the first
formula. Let $G$ be a digraph on a finite set $N$, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(G)$. By Theorem 14, we have

$\displaystyle(-1)^{|N|}S\chi^{O}(G,\mathfrak{G},t,\mathbf{x})$
$\displaystyle=\frac{(-1)^{|N|}}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}S\chi(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum\limits_{\mathfrak{g}\in\mathfrak{G}}\operatorname{sgn}(\mathfrak{g})\overline{\chi}(G,\mathfrak{G},t,\mathbf{x};\mathfrak{g})$
$\displaystyle=\overline{\chi}^{+}(G,\mathfrak{G},t,\mathbf{x}).$

∎

##  9\. Future Directions

First, we note that Stapledon [19] defines a different generalization of the
$h^{\ast}$-vector. His work involves a group $\mathfrak{G}$ acting on the
lattice points of a polytope. Given a quasipolynomial $p(x)$, whose
coefficients are characters, he defines

$\sum\limits_{n\geq 0}p(n)t^{n}=\frac{h^{\ast}(t)}{(1-t)\det[I-t\rho]}$

The $h^{\ast}$-vector is defined for any class function that takes on values
in the ring of quasipolynomials, while our $h$-vector is only defined for
class functions in the ring of polynomials, so the $h^{\ast}$-vector is a more
general invariant. When we restrict to polynomials, the $h^{\ast}$-vector is
different than the $h$-vector. Is the order polynomial class function of a
double poset $h^{\ast}$-effective? Is the chromatic polynomial of a digraph
$h^{\ast}$-effective?

There are other bases of quasisymmetric functions. A very recent basis is the
basis of quasisymmetric power sums $\Psi_{\alpha}$, introduced in [3]. It has
been shown that the $P$-partition enumerator, and the chromatic quasisymmetric
function of a digraph are both $\Psi$-positive [2]. Is the $D$-partition
quasisymmetric class function $\Psi$-effective? Is the chromatic
quasisymmetric class function $\Psi$-effective?

We also would like to have a better description of
$\Omega^{O}(D,\mathfrak{G},\mathbf{x})$ in the $F$ basis. We know the
coefficients are positive. What do they count?

On a similar note, is the $f$-vector $\Omega^{O}(D,\mathfrak{G},x)$
effectively unimodal? This would mean that there exists an $i$ such that
$f_{j}\leq_{\mathfrak{G}}f_{k}$ for $k\leq i$, and
$f_{j}\geq_{\mathfrak{G}}f_{k}$ for $i\leq j$. This is still an open question
even for trivial group actions. Our example in Figure 4 shows that the
$h$-vector for a locally special double poset can fail to be effectively
unimodal. We can obtain the $h$-vector of $\Omega^{O}(D,\mathfrak{G},x)$ by
taking the coefficients of $\chi_{1}$ in the $h$-vector of
$\Omega(D,\mathfrak{G},x)$. Doing so results in the sequence $(0,0,1,0,1,0)$
which fails to be unimodal.

Finally, we could consider proving our results using the theory of
combinatorial Hopf monoids, as studied by Aguiar and Mahajan [1]. We will
pursue this idea in a subsequent paper, using the theory of Hopf monoids to
prove $F$-effectiveness and combinatorial reciprocity results for
quasisymmetric class function invariants associated other combinatorial Hopf
monoids.

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