# Enriched toric $[\vec{D}]$-partitions

Jinting Liang
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
(
Key Words: Cyclic peak, cyclic permutation, cyclic quasi-symmetric function,
enriched $P$-partition, toric poset, order polynomial
AMS subject classification (2010): 05A05 (Primary) 05E05, 06A11 (secondary) )

###### Abstract

This paper develops the theory of enriched toric $[\vec{D}]$-partitions.
Whereas Stembridge’s enriched $P$-partitions give rises to the peak algebra
which is a subring of the ring of quasi-symmetric functions
$\operatorname{QSym}$, our enriched toric $[\vec{D}]$-partitions will generate
the cyclic peak algebra which is a subring of cyclic quasi-symmetric functions
$\operatorname{cQSym}$. In the same manner as the peak set of linear
permutations appears when considering enriched $P$-partitions, the cyclic peak
set of cyclic permutations plays an important role in our theory. The
associated order polynomial is discussed based on this framework.

##  1 Introduction

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of nonnegative integers and
positive integers respectively. For $m,n\in\mathbb{N}$, define
$[m,n]=\\{m,m+1,\ldots,n\\}$ and, as an abbreviation, we write $[n]=[1,n]$
when $m=1$. A linear permutation of a set $A\subset\mathbb{P}$ is an
arrangement $w=w_{1}w_{2}\dots w_{n}$ of elements in $A$ where each element is
used exactly once. In this case, we call $n$ the length of $w$, written as
$\\#w=|w|=n$. Let $\mathcal{S}_{n}$ be the symmetric group on $[n]$ viewed as
linear permutations of $[n]$.

A linear permutation statistic is a function whose domain is the set of all
linear permutations. For a linear permutation $w=w_{1}w_{2}\dots w_{n}$, a
descent of $w$ is a position $i$ such that $w_{i}>w_{i+1}$. The descent set
$\operatorname{Des}$ is defined by

$\operatorname{Des}w=\\{i\mid\text{$i$ is a descent of $w$}\\}\subseteq[n-1].$

The descent number of $w$ is $\operatorname{des}w:=|\operatorname{Des}w|$. A
peak of $w$ is a position $i$ such that $w_{i-1}<w_{i}>w_{i+1}$. The peak set
$\operatorname{Pk}$ is defined by

$\operatorname{Pk}w=\\{i\mid\text{$i$ is a peak of $w$}\\}\subseteq[2,n-1].$

The peak number is $\operatorname{pk}w:=|\operatorname{Pk}w|$.

Quasi-symmetric functions first appeared implicitly as generating functions in
Richard Stanley’s theory of $P$-partitions [Sta72], and then were studied by
Ira M. Gessel [Ges84] in his work. To be more precise, for a finite poset $P$,
the set of $P$-partitions can be partitioned by the linear extensions of $P$,
while each subset corresponds to a fundamental quasi-symmetric function
indexed by the descent set $\operatorname{Des}$ of that linear permutation.
The ring $\operatorname{QSym}$ of quasi-symmetric functions was further
developed, see [BHvW03, GZ18, Sta01] for partial related articles. It also
found applications in enumerative combinatorics, representation theory and
algebraic geometry [Hiv00, ABS06, SW12]. In the same vein, Stembridges’s work
[Ste97] on enriched $P$-partitions gave rise to the algebra of peaks $\Pi$, a
graded subring of $\operatorname{QSym}$, which is closed related to the peak
set $\operatorname{Pk}$.

For a linear permutation $w=w_{1}w_{2}\dots w_{n}$, we define the
corresponding cyclic permutation $[w]$ to be the set of rotations of $w$, that
is,

$[w]=\\{w_{1}w_{2}\dots w_{n},\hskip 7.0ptw_{2}\dots w_{n}w_{1},\hskip
7.0pt\ldots,\hskip 7.0ptw_{n}w_{1}\dots w_{n-1}\\}.$

Let $[\mathcal{S}_{n}]$ denote the set of cyclic permutations on $[n]$.

The cyclic descent set $\operatorname{cDes}$ of a linear permutation $w$ is
defined by

$\operatorname{cDes}w=\\{i\mid\text{$w_{i}>w_{i+1}$ where the subscripts are
taken modulo $n$}\\}\subseteq[n].$

This leads to the cyclic descent set of a cyclic permutation

$\operatorname{cDes}[w]=\\{\\{\operatorname{cDes}\sigma\mid\sigma\in[w]\\}\\},$

where the double curly brackets denote a multiset. The multiplicity comes into
play since $\operatorname{cDes}$ may have the same value on different
representatives $\sigma$ in $[w]$. In fact, one can regard the cyclic
permutation statistic $\operatorname{cDes}$ as an analogy of
$\operatorname{Des}$ in the linear setting. Similarly, if we define the cyclic
peak set $\operatorname{cPk}$ of a linear permutation $w$ by

$\operatorname{cPk}w=\\{i\mid\text{$w_{i-1}<w_{i}>w_{i+1}$ where the
subscripts are taken modulo $n$}\\}\subseteq[n],$

then the cyclic counterpart of $\operatorname{Pk}$, the cyclic peak set
$\operatorname{cPk}$ of a cyclic permutation is defined as

$\operatorname{cPk}[w]=\\{\\{\operatorname{cPk}\sigma\mid\sigma\in[w]\\}\\}.$

######  Example 1.1.

Consider permutation $w=3124$ on $[4]$, then

$\operatorname{Des}w=\\{1\\},\;\operatorname{cDes}w=\\{1,4\\},\;\operatorname{Pk}w=\emptyset,\;\operatorname{cPk}w=\\{4\\}.$

######  Remark 1.2.

By definition, $\operatorname{cDes}[w]$ carries the information for all
representatives in $[w]$. Moreover, $\operatorname{cDes}w$ together with $|w|$
will be sufficient to determine $\operatorname{cDes}[w]$. In fact,
$\operatorname{cDes}[w]$ is simply collecting all cyclic shifts of
$\operatorname{cDes}w$ in $[n]$ where $n=|w|$, namely,

$\operatorname{cDes}[w]=\\{\\{\,i+\operatorname{cDes}w\mid i\in[n]\,\\}\\}.$

Here $i+\operatorname{cDes}w$ is the set defined by (2.1). Similarly,
$\operatorname{cPk}[w]$ can be entirely determined by $\operatorname{cPk}w$
and $|w|$.

######  Example 1.3.

Consider the permutation $w=1423$ and the corresponding cyclic permutation
$[w]=\\{1423,3142,2314,4231\\}$, we have

$\operatorname{cDes}[w]=\operatorname{cPk}[w]=\\{\\{\,\\{2,4\\},\\{1,3\\},\\{2,4\\},\\{1,3\\}\,\\}\\}.$

Noting that $\operatorname{cDes}w=\operatorname{cPk}w=\\{2,4\\}$, one can
easily check the previous remark does hold.

In the work [AGRR20] of Adin, Gessel, Reiner, and Roichman, the ring
$\operatorname{cQSym}$ of cyclic quasi-symmetric functions was introduced from
toric $P$-partition enumerators, in which case the cyclic descent set
$\operatorname{cDes}$ plays an important role. The authors also asked for a
cyclic version of the algebra of peaks, to which question we will give an
answer in this paper.

This article is devoted to the study of enriched toric $[\vec{D}]$-partitions.
By the end, we will construct an algebra of cyclic peaks in
$\operatorname{cQSym}$ analogously as the algebra of peaks. The rest of this
paper is structured as follows. In the next section, we recall definitions of
various terms such as quasi-symmetric functions and cyclic quasi-symmetric
functions, with several concrete examples provided. Section 3 introduces
enriched $\vec{D}$-partitions in terms of directed acyclic graphs (DAGs). In
section 4, we define enriched toric $[\vec{D}]$-partitions and develop some of
their properties. Section 5 will review the weight enumerators of enriched
$\vec{D}$-partitions from Stembridge and discuss the cyclic analogies for
enriched toric $[\vec{D}]$-partitions. Moreover, the weight enumerators
corresponding to different cyclic peak sets generate a subring of
$\operatorname{cQSym}$ and we call it the algebra of cyclic peaks. We also
compute the order polynomial of enriched toric $[\vec{D}]$-partitions.

##  2 Basic definitions and results

###  2.1 Sets and compositions

We will use $\leq$ with no subscript to denote the ordinary total order on
$\mathbb{Z}$, the set of integers. For $n\in\mathbb{P}$, let $2^{[n]}$ denote
the set of all subsets of $[n]$, and $2^{[n]}_{0}$ be the set of all nonempty
subsets of $[n]$. Denote by $\operatorname{Comp}_{n}$ the set of all
compositions of $n$ and write $\alpha\vDash n$ for
$\alpha\in\operatorname{Comp}_{n}$. Define a cyclic shift of a subset
$E\subseteq[n]$ in $[n]$ to be a set of the form

$i+E=\\{\text{$i+e$ (mod n)}\mid e\in E\\}.$ (2.1)

Note that sometimes we will use $E+i$ as well for the same concept. While
using a negative shift, the reader should be careful to distinguish between
$E-i$ and the set difference $E-\\{i\\}=E\setminus\\{i\\}$.

A cyclic shift of a composition
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{m})$ is a composition of the
form

$(\alpha_{k},\ldots,\alpha_{m},\alpha_{1},\ldots,\alpha_{k-1})$

for some $k\in[m]$. We adopt the notations from [AGRR20] and denote by
$c2^{[n]}_{0}$ (respectively, $\operatorname{cComp}_{n}$) the set of
equivalence classes of elements of $2^{[n]}_{0}$ (respectively,
$\operatorname{Comp}_{n}$) under cyclic shifts. Here we recall two natural
bijections which will play important roles when indexing two particular bases
of (cyclic) quasi-symmetric functions.

The first natural bijection is between $2^{[n-1]}$ and
$\operatorname{Comp}_{n}$. The map $\Phi:2^{[n-1]}\to\operatorname{Comp}_{n}$
is defined by

$\Phi(E):=(e_{1}-e_{0},e_{2}-e_{1},\ldots,e_{k}-e_{k-1},e_{k+1}-e_{k})$ (2.2)

for any given $E=\\{e_{1}<e_{2}<\dots<e_{k}\\}\subseteq[n-1]$ with $e_{0}=0$
and $e_{k+1}=n$, where the inverse map is

$\Phi^{-1}(\alpha)=\\{\alpha_{1},\alpha_{1}+\alpha_{2},\ldots,\alpha_{1}+\alpha_{2}+\dots+\alpha_{k}\\}$

for any $\alpha=(\alpha_{1},\ldots,\alpha_{k+1})\vDash n$.

Another bijection is between $c2_{0}^{[n]}$ and $\operatorname{cComp}_{n}$,
for the sake of which we need to consider the map
$\psi:2_{0}^{[n]}\to\operatorname{Comp}_{n}$ defined by

$\psi(E):=(e_{2}-e_{1},\ldots,e_{k}-e_{k-1},e_{1}-e_{k}+n)$ (2.3)

where $E=\\{e_{1}<e_{2}<\dots<e_{k}\\}\subseteq[n]$. Notice that if
$E^{\prime}$ is a cyclic shift of $E$ in $[n]$, then $\psi(E^{\prime})$ is
also a cyclic shift of $\psi(E)$. Therefore $\psi$ induces a map
$\Psi:c2_{0}^{[n]}\to\operatorname{cComp}_{n}$. Moreover, it is
straightforward to check that the induced map $\Psi$ is bijective.

###  2.2 Quasi-symmetric functions $\operatorname{QSym}$

A quasi-symmetric function is a formal power series
$f\in\mathbb{Q}[[x_{1},x_{2},\ldots]]$ such that for any sequence of positive
integers $a=(a_{1},a_{2},\ldots,a_{s})$, and two increasing sequences
$i_{1}<i_{2}<\dots<i_{s}$ and $j_{1}<j_{2}<\dots<j_{s}$ of positive integers,

$[x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\dots
x_{i_{s}}^{a_{s}}]\,f=[x_{j_{1}}^{a_{1}}x_{j_{2}}^{a_{2}}\dots
x_{j_{s}}^{a_{s}}]\,f,$

where $[x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\dots x_{i_{s}}^{a_{s}}]\,f$ denotes
the coefficient of monomial $x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\dots
x_{i_{s}}^{a_{s}}$ in the expression of $f$. Let $\operatorname{QSym}_{n}$ be
the set of all quasi-symmetric functions which are homogeneous of degree $n$,
and $\operatorname{QSym}=\oplus_{n\geq 0}\operatorname{QSym}_{n}$. Two bases
of $\operatorname{QSym}$ are particularly important to our work: monomial
quasi-symmetric functions $M_{L}$ and fundamental quasi-symmetric functions
$F_{L}$.

Given a composition $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{s})\vDash
n$, the associated monomial quasi-symmetric function indexed by $\alpha$ is

$M_{\alpha}=\sum_{i_{1}<i_{2}<\dots<i_{s}}x_{i_{1}}^{\alpha_{1}}x_{i_{2}}^{\alpha_{2}}\dots
x_{i_{s}}^{\alpha_{s}}.$

It is clear that $\\{M_{\alpha}\\}_{\alpha\vDash n}$ form a basis of
$\operatorname{QSym}_{n}$. From the natural bijection
$\Phi:2^{[n-1]}\to\operatorname{Comp}_{n}$ defined by (2.2), we can also index
the monomial quasi-symmetric functions by subsets $E\subseteq[n-1]$, and
define $M_{n,E}:=M_{\Phi(E)}$.

There is another important basis of $\operatorname{QSym}$. The fundamental
quasi-symmetric function indexed by $E\subseteq[n-1]$ is

$F_{n,E}=\sum_{\begin{subarray}{c}i_{1}\leq\dots\leq i_{n}\\\\[4.0pt]
i_{k}<i_{k+1}\,\text{if $k\in E$}\end{subarray}}x_{i_{1}}x_{i_{2}}\dots
x_{i_{n}}.$

Similarly, we can define $F_{\alpha}:=F_{\Phi^{-1}(\alpha)}$ indexed by
compositions.

The relation between monomial and fundamental quasi-symmetric functions is
simple:

$F_{n,E}=\sum_{L\supseteq E}M_{n,L}.$ (2.4)

By the principle of inclusion and exclusion, $M_{n,E}$ can be expressed as a
linear combination of the $F_{n,L}$, from which we can tell that
$\\{F_{n,L}\\}_{L\subseteq[n-1]}$ spans $\operatorname{QSym}_{n}$. By checking
the cardinality of both sets $\\{F_{n,L}\\}_{L\subseteq[n-1]}$ and
$\\{M_{n,E}\\}_{E\subseteq[n-1]}$, it follows that
$\\{F_{n,L}\\}_{L\subseteq[n-1]}$ is indeed a basis of
$\operatorname{QSym}_{n}$.

######  Example 2.1.

Consider $E=\\{1,3\\}$, by definition we have

$F_{4,\\{1,3\\}}=\sum_{i_{1}<i_{2}\leq
i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}=\sum\limits_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}+\sum_{i_{1}<i_{2}<i_{4}}x_{i_{1}}x_{i_{2}}^{2}x_{i_{4}}.$

There are only two choices for a set $L$ satisfying that $E\subseteq
L\subseteq[3]$: $\\{1,3\\}$ or $\\{1,2,3\\}$. Since
$\Phi(\\{1,3\\})=(1,2,1),\Phi(\\{1,2,3\\})=(1,1,1,1)\,$, we get

$M_{4,\\{1,3\\}}=M_{(1,2,1)}=\sum_{i_{1}<i_{2}<i_{3}}x_{i_{1}}x_{i_{2}}^{2}x_{i_{3}},\hskip
5.0ptM_{4,\\{1,2,3\\}}=M_{(1,1,1,1)}=\sum_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}.$

The calculation above verifies that
$F_{4,\\{1,3\\}}=M_{4,\\{1,3\\}}+M_{4,\\{1,2,3\\}}=\sum\limits_{L\supseteq\\{1,3\\}}M_{4,L}$.

###  2.3 Cyclic quasi-symmetric functions $\operatorname{cQSym}$

In this subsection, we recall from [AGRR20] the theory of cyclic quasi-
symmetric functions. We will model our work of enriched toric
$[\vec{D}]$-partitions with enumerators in this environment.

A cyclic quasi-symmetric function is a formal power series
$f\in\mathbb{Q}[[x_{1},x_{2},\ldots]]$ such that for any sequence of positive
integers $a=(a_{1},a_{2},\ldots,a_{s})$, a cyclic shift
$(a^{\prime}_{1},a^{\prime}_{2},\ldots,a^{\prime}_{s})$ of $a$, and two
increasing sequences $i_{1}<i_{2}<\dots<i_{s}$ and $j_{1}<j_{2}<\dots<j_{s}$
of positive integers,

$[x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\dots
x_{i_{s}}^{a_{s}}]\,f=[x_{j_{1}}^{a^{\prime}_{1}}x_{j_{2}}^{a^{\prime}_{2}}\dots
x_{j_{s}}^{a^{\prime}_{s}}]\,f,$

namely the coefficients of $x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\dots
x_{i_{s}}^{a_{s}}$ and
$x_{j_{1}}^{a^{\prime}_{1}}x_{j_{2}}^{a^{\prime}_{2}}\dots
x_{j_{s}}^{a^{\prime}_{s}}$ in $f$ are equal. Denote by
$\operatorname{cQSym}_{n}$ the set of all cyclic quasi-symmetric functions
which are homogeneous of degree $n$, and $\operatorname{cQSym}=\oplus_{n\geq
0}\operatorname{cQSym}_{n}$.

######  Remark 2.2.

It is clear that there exists a strict inclusion relation
$\operatorname{Sym}\subsetneq\operatorname{cQSym}\subsetneq\operatorname{QSym}$,
where $\operatorname{Sym}$ is the algebra of symmetric functions.

We have the following cyclic analogues of the concepts of monomial
(fundamental) quasi-symmetric functions.

Given a composition $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{s})\vDash
n$, the associated monomial cyclic quasi-symmetric function indexed by
$\alpha$ is

$M^{\operatorname{cyc}}_{\alpha}=\sum_{i=1}^{s}M_{(\alpha_{i},\alpha_{i+1},\ldots,\alpha_{i-1})},$

where the indices are interpreted modulo $s$, meaning
$\alpha_{j}=\alpha_{j+s}$. In other words, $M^{\operatorname{cyc}}_{\alpha}$
sums over all monomial quasi-symmetric functions indexed by cyclic shifts of
$\alpha$. Therefore it is clear that
$M^{\operatorname{cyc}}_{\alpha}=M^{\operatorname{cyc}}_{\alpha^{\prime}}$ if
$\alpha$ and $\alpha^{\prime}$ only differ by a cyclic shift.

We can also index the monomial cyclic quasi-symmetric function by sets. For a
nonempty $E\subseteq[n]$, define
$M^{\operatorname{cyc}}_{n,E}:=M^{\operatorname{cyc}}_{\psi(E)}$ via the map
$\psi:2_{0}^{[n]}\to\operatorname{Comp}_{n}$ defined by (2.3), and set
$M^{\operatorname{cyc}}_{n,\emptyset}:=0$. Similarly it can be shown that
$M^{\operatorname{cyc}}_{n,E}=M^{\operatorname{cyc}}_{n,E^{\prime}}$ if
$E^{\prime}$ is a cyclic shift of $E$.

The following result gives the expression of monomial cyclic quasi-symmetric
functions in terms of monimial quasi-symmetric functions.

######  Lemma 2.3 ([AGRR20] Lemma 2.5, monomial to cyclic monomial).

For any subset $E\subseteq[n]$

$M^{\operatorname{cyc}}_{n,E}=\sum_{e\in E}M_{n,(E-e)\cap[n-1]},$ (2.5)

where the set $E-e$ is defined as (2.1). ∎

######  Example 2.4.

Table 1 computes all monomial cyclic quasi-symmetric function indexed by
compositions of $4$, in terms of monomial quasi-symmetric functions.

Table 1: Monomial cyclic quasi-symmetric functions indexed by compositions of
$4$ $\begin{array}[]{|c|c|}\hline\cr\alpha\vDash
4&M^{\operatorname{cyc}}_{\alpha}\\\
\hline\cr\hline\cr(4)&M_{(4)}=M_{4,\emptyset}\\\ \hline\cr\text{(1,3) or
(3,1)}&M_{(1,3)}+M_{(3,1)}=M_{4,\\{1\\}}+M_{4,\\{3\\}}\\\
\hline\cr(2,2)&2M_{(2,2)}=2M_{4,\\{2\\}}\\\ \hline\cr\text{(1,1,2) or (1,2,1)
or
(2,1,1)}&M_{(1,1,2)}+M_{(1,2,1)}+M_{(2,1,1)}=M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}}\\\
\hline\cr(1,1,1,1)&4M_{(1,1,1,1)}=4M_{4,\\{1,2,3\\}}\\\ \hline\cr\end{array}$

For the natural desire of establishing a similar relation as the one between
monomial and fundamental quasi-symmetric functions given by (2.4) in our
cyclic situation, define the fundamental cyclic quasi-symmetric function
indexed by $E\subseteq[n]$ as

$F_{n,E}^{\operatorname{cyc}}:=\sum_{L\supseteq
E}M_{n,L}^{\operatorname{cyc}}.$ (2.6)

######  Remark 2.5.

This is not the original definition in [AGRR20] but appears as a lemma in the
same article. But these two definitions are equivalent via [AGRR20] Lemma
2.14. We will use this definition for the purpose of our work, and mention the
original definition in Proposition 2.11 for interested readers.

######  Example 2.6.

Consider $n=4$ and $E=\\{1,3\\}$. By definition

$\displaystyle F_{4,\\{1,3\\}}^{\operatorname{cyc}}=$
$\displaystyle\quad\sum_{L\supseteq\\{1,3\\}}M_{4,L}^{\operatorname{cyc}}$
$\displaystyle\overset{(i)}{=}$ $\displaystyle\quad
M_{4,\\{1,3\\}}^{\operatorname{cyc}}+M_{4,\\{1,2,3\\}}^{\operatorname{cyc}}+M_{4,\\{1,3,4\\}}^{\operatorname{cyc}}+M_{4,\\{1,2,3,4\\}}^{\operatorname{cyc}}$
$\displaystyle\overset{(ii)}{=}$ $\displaystyle\quad
M_{(2,2)}^{\operatorname{cyc}}+M_{(1,1,2)}^{\operatorname{cyc}}+M_{(2,1,1)}^{\operatorname{cyc}}+M_{(1,1,1,1)}^{\operatorname{cyc}}$
$\displaystyle\overset{(iii)}{=}$ $\displaystyle\quad
2M_{(2,2)}+2\left(M_{(1,1,2)}+M_{(1,2,1)}+M_{(2,1,1)})\right)+4M_{(1,1,1,1)}$
$\displaystyle=$ $\displaystyle\quad
2\sum_{i_{1}<i_{2}}x_{i_{1}}^{2}x_{i_{2}}^{2}$
$\displaystyle\quad+2\sum_{i_{1}<i_{2}<i_{3}}(x_{i_{1}}x_{i_{2}}x_{i_{3}}^{2}+x_{i_{1}}x_{i_{2}}^{2}x_{i_{3}}+x_{i_{1}}^{2}x_{i_{2}}x_{i_{3}})$
$\displaystyle\quad+4\sum_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}.$

Equality $(i)$ follows from the fact that the choices for
$L\supseteq\\{1,3\\}$ in $[4]$ are $\\{1,3\\}$, $\\{1,2,3\\}$, $\\{1,3,4\\}$
and $\\{1,2,3,4\\}$. Equality $(ii)$ is obtained by changing indices under the
map $\psi$ defined by (2.3), equality (iii) is from Table 1.

The following passing from fundamental to cyclic fundamental quasi-symmetric
functions should come without surprise.

######  Lemma 2.7 ([AGRR20] Proposition 2.15, fundamental to cyclic
fundamental).

For any subset $E\subseteq[n]$,

$F^{\operatorname{cyc}}_{n,E}=\sum_{i\in[n]}F_{n,(E-i)\cap[n-1]},$

with set $E-i$ defined by (2.1). ∎

######  Remark 2.8.

  1. 1.

It follows directly from Lemma 2.7 that
$F^{\operatorname{cyc}}_{n,E}=F^{\operatorname{cyc}}_{n,E^{\prime}}$ if
$E^{\prime}$ is a cyclic shift of $E$.

  2. 2.

Clearly the set $\\{M^{\operatorname{cyc}}_{n,E}:E\in c2^{[n]}_{0}\\}$ spans
$\operatorname{cQSym}_{n}$ and is linearly independent, as each monomial of
degree $n$ appears in $M^{\operatorname{cyc}}_{n,E}$ for exactly one $E\in
c2^{[n]}_{0}$. Hence $\\{M^{\operatorname{cyc}}_{n,E}:E\in c2^{[n]}_{0}\\}$ is
a basis of $\operatorname{cQSym}_{n}$. Applying the principle of inclusion and
exclusion on (2.6) we have

$M^{\operatorname{cyc}}_{n,E}=\sum_{L\supseteq E}(-1)^{|L\setminus
E|}F^{\operatorname{cyc}}_{n,L},$

which implies that $\\{F^{\operatorname{cyc}}_{n,E}:E\in c2^{[n]}_{0}\\}$ also
spans $\operatorname{cQSym}_{n}$, together with the fact that the dimension of
vector space $\operatorname{cQSym}_{n}$ is $\\#c2^{[n]}_{0}$,
$\\{F^{\operatorname{cyc}}_{n,E}:E\in c2^{[n]}_{0}\\}$ is also a basis of
$\operatorname{cQSym}_{n}$.

Now we review the original definition of cyclic quasi-symmetric functions.
Before that, we need to associate each $E\subseteq[n]$ with a set
$P_{n,E}^{\operatorname{cyc}}$.

######  Definition 2.9.

Let $P_{n,E}^{\operatorname{cyc}}$ denote the set of all pairs $(w,k)$ where
$w=w_{1}\ldots w_{n}$ a sequence of positive integers and $k\in[n]$ satisfying

  1. (1)

$w$ is “cyclically weakly increasing” from index $k$, that is, $w_{k}\leq
w_{k+1}\leq\ldots\leq w_{n}\leq w_{1}\leq\ldots\leq w_{k-1}$.

  2. (2)

If $i\in E\setminus\\{k-1\\}$, then $w_{i}<w_{i+1}$, where the indices are
computed modulo $n$.

######  Remark 2.10.

The index $k$ is uniquely defined by $w$ unless all integers in $w$ are the
same. In that case, it must be either $E=\\{k-1\\}$ for some $k\in[n]$ or
$E=\emptyset$. Those $w$ with elements all equal will only be paired with
$k-1$ if $E=\\{k-1\\}$, and so get counted once in
$P_{n,E}^{\operatorname{cyc}}$; as for $E=\emptyset$, if $w$ has all elements
equal, it can pair with every $k\in[n]$, therefore it will be counted $n$
times in $P_{n,E}^{\operatorname{cyc}}$.

######  Proposition 2.11.

For every subset $E\subseteq[n]$,

$F_{n,E}^{\operatorname{cyc}}=\sum_{(w,k)\in
P_{n,E}^{\operatorname{cyc}}}x_{w_{1}}x_{w_{2}}\dots x_{w_{n}}.$

###### Proof.

Despite the statement, the proof is essentially the same as for [AGRR20] Lemma
2.14. ∎

Here we provide a concrete example to illustrate the validity of the
proposition above.

######  Example 2.12.

Consider $n=4$ and $E=\\{1,3\\}$. It is clear that
$P_{n,E}^{\operatorname{cyc}}$ has a natural partition into four parts where
each part contains all pairs with a given index $k\in[4]$. This implies the
following summation

$\displaystyle\sum_{(w,k)\in
P_{4,\\{1,3\\}}^{\operatorname{cyc}}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}$
$\displaystyle=\sum_{k=1}^{4}\sum_{(w,k)\in
P_{4,\\{1,3\\}}^{\operatorname{cyc}}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}$
$\displaystyle=\sum_{w_{1}<w_{2}\leq
w_{3}<w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}+\sum_{w_{2}\leq
w_{3}<w_{4}\leq w_{1}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}$
$\displaystyle\qquad+\sum_{w_{3}<w_{4}\leq
w_{1}<w_{2}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}+\sum_{w_{4}\leq
w_{1}<w_{2}\leq w_{3}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}},$

For each summand,

$\displaystyle\begin{split}\sum_{w_{1}<w_{2}\leq
w_{3}<w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}&=\sum_{w_{1}<w_{2}<w_{3}<w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}+\sum_{w_{1}<w_{2}=w_{3}<w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}\\\
&=\sum_{w_{1}<w_{2}<w_{3}<w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}+\sum_{w_{1}<w_{2}<w_{3}}x_{w_{1}}x_{w_{2}}^{2}x_{w_{3}};\end{split}$
$\displaystyle\begin{split}\sum_{w_{2}\leq w_{3}<w_{4}\leq
w_{1}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}&=\sum_{w_{2}<w_{3}<w_{4}<w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}x_{w_{1}}+\sum_{w_{2}=w_{3}<w_{4}<w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}x_{w_{1}}\\\
&\qquad+\sum_{w_{2}<w_{3}<w_{4}=w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}x_{w_{1}}+\sum_{w_{2}=w_{3}<w_{4}=w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}x_{w_{1}}\\\
&=\sum_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}+\sum_{i_{1}<i_{2}<i_{3}}x_{i_{1}}^{2}x_{i_{2}}x_{i_{3}}+\sum_{i_{1}<i_{2}<i_{3}}x_{i_{1}}x_{i_{2}}x_{i_{3}}^{2}+\sum_{i_{1}<i_{2}}x_{i_{1}}^{2}x_{i_{2}}^{2};\end{split}$
$\displaystyle\begin{split}\sum_{w_{3}<w_{4}\leq
w_{1}<w_{2}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}&=\sum_{w_{3}<w_{4}<w_{1}<w_{2}}x_{w_{3}}x_{w_{4}}x_{w_{1}}x_{w_{2}}+\sum_{w_{3}<w_{4}=w_{1}<w_{2}}x_{w_{3}}x_{w_{4}}x_{w_{1}}x_{w_{2}}\\\
&=\sum_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}+\sum_{w_{1}<w_{2}<w_{3}}x_{w_{1}}x_{w_{2}}^{2}x_{w_{3}};\end{split}$
$\displaystyle\begin{split}\sum_{w_{4}\leq w_{1}<w_{2}\leq
w_{3}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}&=\sum_{w_{4}<w_{1}<w_{2}<w_{3}}x_{w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}+\sum_{w_{4}=w_{1}<w_{2}<w_{3}}x_{w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}\\\
&\qquad+\sum_{w_{4}<w_{1}<w_{2}=w_{3}}x_{w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}+\sum_{w_{4}=w_{1}<w_{2}=w_{3}}x_{w_{4}}x_{w_{1}}x_{w_{2}}x_{w_{3}}\\\
&=\sum_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}+\sum_{i_{1}<i_{2}<i_{3}}x_{i_{1}}^{2}x_{i_{2}}x_{i_{3}}+\sum_{i_{1}<i_{2}<i_{3}}x_{i_{1}}x_{i_{2}}x_{i_{3}}^{2}+\sum_{i_{1}<i_{2}}x_{i_{1}}^{2}x_{i_{2}}^{2}.\end{split}$

To sum up, we have

$\displaystyle\sum_{(w,k)\in
P_{4,\\{1,3\\}}^{\operatorname{cyc}}}x_{w_{1}}x_{w_{2}}x_{w_{3}}x_{w_{4}}$
$\displaystyle=4\sum_{i_{1}<i_{2}<i_{3}<i_{4}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}$
(2.7)
$\displaystyle\qquad+2\sum_{i_{1}<i_{2}<i_{3}}(x_{i_{1}}^{2}x_{i_{2}}x_{i_{3}}+x_{i_{1}}x_{i_{2}}^{2}x_{i_{3}}+x_{i_{1}}x_{i_{2}}x_{i_{3}}^{2})$
$\displaystyle\qquad+2\sum_{i_{1}<i_{2}}x_{i_{1}}^{2}x_{i_{2}}^{2},$

which is exactly the expression of $F_{4,\\{1,3\\}}^{\operatorname{cyc}}$ we
obtained in Example 2.6.

##  3 Enriched $\vec{D}$-partitions for DAGs

In [Ste97], Stembridge defined the enriched $P$-partition of a poset $P$.
Different from having $\mathbb{P}$ with the ordinary order $\leq$ as range for
ordinary $P$-partitions, enriched $P$-partitions are obtained by imposing
another total order on the range, $\mathbb{P^{\prime}}$, which is the set of
nonzero integers in an unusual order. We review the basic theory from
Stembridge and naturally extend enriched $P$-partitions to enriched
$\vec{D}$-partitions where $\vec{D}$ is a directed acyclic graph which is not
necessarily transitive.

A directed acyclic graph (DAG) is a digraph with no directed cycles. Suppose
$\vec{D}$ is a DAG with vertex set $[n]$. A DAG $\vec{D}$ is transitive if
$i\to j$ and $j\to k$ implies $i\to k$ for $i,j,k\in[n]$. If $\vec{P}$ is a
transitive DAG, it will naturally induce a partial order $<_{\vec{P}}$ on the
vertex set $[n]$, defined so that for two vertices $i$ and $j$, one has
$i<_{\vec{P}}j$ if and only if $i\to j$ in $\vec{P}$; in that case, we also
use $\vec{P}$ to denote this poset.

In general, we can associate a partial order with any DAG $\vec{D}$. For this
purpose, define the transitive closure $\vec{P}$ of $\vec{D}$ as the directed
graph obtained from $\vec{D}$ by adding in $i\to k$ if one has both $i\to j$
and $j\to k$ in $\vec{D}$. Such $\vec{P}$ is unique. Moreover, it is
straightforward to verify that the transitive closure $\vec{P}$ is both
acyclic and transitive, where the acyclicity is ensured by the acyclic
condition of $\vec{D}$ and the transitivity comes from the added conditions.
This implies that $\vec{P}$ is actually a transitive DAG, hence has a partial
order structure. We will maintain the use $\to$ for the relation between
vertices in a general DAG $\vec{D}$, while using the partial order
$\leq_{\vec{P}}$ on a poset $\vec{P}$.

A poset $\vec{P}$ is a total linear order if it is a complete DAG, i.e., there
is a directed edge between every pair of vertices in $\vec{P}$. There is a
bijection between the set of total linear orders $\vec{P}$ on the vertex set
$[n]$ and $\mathcal{S}_{n}$. For a total linear order $\vec{P}$ on $[n]$,
there exists a unique directed path $w_{1}\to w_{2}\to\dots\to w_{n}$ in
$\vec{P}$, hence $\vec{P}$ can be identified with the permutation
$w=w_{1}w_{2}\dots w_{n}\in\mathcal{S}_{n}$. Conversely, given a permutation
$w=w_{1}w_{2}\dots w_{n}\in\mathcal{S}_{n}$, we can construct a DAG $\vec{P}$
by putting arrows $w_{i}\to w_{j}$ for all $1\leq i<j\leq n$ on the vertex set
$[n]$, and it is easy to check that the resulting DAG $\vec{P}$ is actually a
total linear order. In this case, we usually use a permutation $w$ to denote
the corresponding total linear order $\vec{P}$.

For two DAGs $\vec{D}_{1}$ and $\vec{D}_{2}$ on the same vertex set $[n]$, we
say $\vec{D}_{2}$ extends $\vec{D}_{1}$ if $\vec{D}_{1}$ is a subgraph of
$\vec{D}_{2}$, written as $\vec{D}_{1}\subseteq\vec{D}_{2}$. If, furthermore,
$\vec{D}_{2}$ is a total linear order corresponding to the permutation
$w\in\mathcal{S}_{n}$, we also say that $w$ linearly extends $\vec{D}_{1}$.
Denote by $\mathcal{L}(\vec{D})$ the set of all permutations
$w\in\mathcal{S}_{n}$ which linearly extend $\vec{D}$.

######  Example 3.1.

Here are several DAGs on the vertex set $[4]=\\{1,2,3,4\\}$.

$\vec{D}_{1}=2431$$2$$4$$3$$1$$\vec{D}_{2}=2413$$2$$4$$3$$1$$\vec{D}_{3}$$2$$4$$3$$1$

Note that $\vec{D}_{1}$ is the total linear order with permutation $2431$,
$\vec{D}_{2}$ is the total linear order for $2413$. Both $\vec{D}_{1}$ and
$\vec{D}_{2}$ extend $\vec{D}_{3}$, hence $2431$ and $2413$ linearly extend
$\vec{D}_{3}$. Moreover, $\vec{D}_{1}$ and $\vec{D}_{2}$ are the only total
linear orders which extend $\vec{D}_{3}$, therefore
$\mathcal{L}(\vec{D}_{3})=\\{2431,2413\\}$.

Now we are in a good position to define enriched $\vec{D}$-partitions for a
DAG $\vec{D}$. Stembridge originally defined the enriched $P$-partitions when
$P$ is a poset. This definition can be easily extended to the cases when
$\vec{D}$ is simply a DAG, as the definition does not rely on the transitivity
of $P$.

Stembridge defines $\mathbb{P}^{\prime}$ to be the set of nonzero integers,
totally ordered as

$-1\prec 1\prec-2\prec 2\prec-3\prec 3\prec\cdots.$

######  Definition 3.2 (Enriched $\vec{D}$-partition).

Let $\vec{D}$ be a directed acyclic graph (DAG) on $[n]$. An enriched
$\vec{D}$-partition is a function $f:[n]\to\mathbb{P^{\prime}}$ such that for
all $i\to j$ in $\vec{D}$,

  1. (a)

$f(i)\preceq f(j)$,

  2. (b)

$f(i)=f(j)>0$ implies $i<j$,

  3. (c)

$f(i)=f(j)<0$ implies $i>j$.

Denote by $\mathcal{E}(\vec{D})$ the set of all enriched $\vec{D}$-partitions
$f$.

######  Remark 3.3.

  1. 1.

In this definition we are using two order structures on the domain $[n]$: the
order $\to$ induced by DAG ${\vec{D}}$ and the ordinary total order $\leq$ on
integers in (b) and (c). Both of them will impose restrictions on the possible
choices for $f$. As for the range $\mathbb{P^{\prime}}$, we also use two order
structures: the total order $\preceq$ defined by Stembridge in (a) and the
usual order $\leq$ on the integers in (b) and (c).

  2. 2.

If $\vec{D}=w$ is a total linear order, the structure of the set of enriched
$w$-partitions is quite simple:

$\begin{array}[]{rl}\mathcal{E}(w)=\\{\,f:[n]\to\mathbb{P^{\prime}}~{}|&f(w_{1})\preceq\cdots\preceq
f(w_{n}),\\\ &f(w_{i})=f(w_{i+1})>0\Rightarrow
i\notin\operatorname{Des}(w),\\\ &f(w_{i})=f(w_{i+1})<0\Rightarrow
i\in\operatorname{Des}(w)\,\\}.\\\ \end{array}$ (3.1)

The following fundamental lemma is a straightforward analogue of Stembridge
[Ste97], Lemma 2.1.

######  Lemma 3.4 (Fundamental lemma of enriched $\vec{D}$-partitions).

For any DAG $\vec{D}$ with vertex set $[n]$, one has a decomposition of
$\mathcal{E}(\vec{D})$ as the following disjoint union:

$\mathcal{E}(\vec{D})=\bigsqcup_{w\in\mathcal{L}(\vec{D})}\mathcal{E}(w).$

###### Proof.

Given an enriched $\vec{D}$-partition $f$. First we arrange the elements of
$[n]$ in a weakly increasing order of $f$-values with respect to the total
order $\preceq$ on the range. Then if some elements in $[n]$ have the same
$f$-value $-k$ (respectively, $+k$) for some positive integer $k$, we arrange
them in a decreasing (respectively, increasing) order with respect to the
usual order $\leq$ on the domain. The resulting permutation $w$ is unique with
$f\in\mathcal{E}(w)$, and $w$ linearly extends $\vec{D}$. On the other hand,
for $w\in\mathcal{L}(\vec{D})$, every enriched $w$-partition is also an
enriched $\vec{D}$-partition. Therefore the conclusion follows. ∎

######  Example 3.5.

Returning to Example 3.1, since $\mathcal{L}(\vec{D}_{3})=\\{2431,2413\\}$, by
the Fundamental Lemma,
$\mathcal{E}(\vec{D}_{3})=\mathcal{E}(2431)\uplus\mathcal{E}(2413)$.

##  4 Enriched toric $[\vec{D}]$-partitions for toric DAGs

In this section, we review the toric DAGs and toric posets as cyclic analogues
of DAGs and posets. Then we define enriched toric $[\vec{D}]$-partitions and
develop some of their properties. The concept of toric poset was originally
defined and studied by Develin, Macauley and Reiner in [DMR16]. Here we follow
the presentation from Adin, Gessel, Reiner and Roichman [AGRR20].

Just like a linear permutation has a corresponding cyclic permutation as the
equivalence class under the equivalence of rotation, for a DAG, we will define
an equivalence relation and consider the equivalence class to be the
corresponding toric DAG. It turns out that if $w$ is a linear extension of the
DAG $\vec{D}$, then $[w]$ is a toric extension of the corresponding toric DAG
$[\vec{D}]$.

A DAG $\vec{D}$ on $[n]$ has $i_{0}\in[n]$ as a source (respectively, sink) if
$\vec{D}$ does not contain $j\to i_{0}$ (respectively, $i_{0}\to j$) for any
$j\in[n]$. Suppose $i_{0}$ is a source or a sink in $\vec{D}$, we say
$\vec{D}^{\prime}$ is obtained from $\vec{D}$ by a flip at $i_{0}$ if
$\vec{D}^{\prime}$ is obtained by reversing all arrows containing $i_{0}$. We
define the equivalence relation $\equiv$ on DAGs as follows:
$\vec{D}^{\prime}\equiv\vec{D}$ if and only if $\vec{D}^{\prime}$ is obtained
from $\vec{D}$ by a sequence of flips. A toric DAG is the equivalence class
$[\vec{D}]$ of a DAG $\vec{D}$.

In particular, if $\vec{D}=w=w_{1}w_{2}\ldots w_{n}$ is a total linear order,
the next proposition claims that the corresponding toric DAG $[\vec{D}]$ can
be identified with the cyclic permutation $[w]$.

######  Proposition 4.1 ([DMR16], Proposition 4.2).

If $\vec{D}=w$ is a total linear order with $w=w_{1}\ldots w_{n}$, then there
is a bijection between toric DAG $[\vec{D}]$ and cyclic permutation $[w]$.∎

######  Example 4.2.

The total linear order $\vec{D}_{1}=2431$ from Example 3.1 has a corresponding
toric DAG $[\vec{D}_{1}]$:

$\vec{D}_{1}=2431$$2$$4$$3$$1$$\vec{D}_{1}^{\prime}=1243$$2$$4$$3$$1$$\vec{D}_{1}^{\prime\prime}=3124$$2$$4$$3$$1$$\vec{D}_{1}^{\prime\prime\prime}=4312$$2$$4$$3$$1$

They can be obtained by a sequence of flips:

$\vec{D}_{1}\xrightarrow{\text{flip at
1}}\vec{D}_{1}^{\prime}\xrightarrow{\text{flip at
3}}\vec{D}_{1}^{\prime\prime}\xrightarrow{\text{flip at
4}}\vec{D}_{1}^{\prime\prime\prime}\xrightarrow{\text{flip at 2}}\vec{D}_{1}.$

Therefore it is easy to see that $[\vec{D}_{1}]$ can be identified with
$[2431]=\\{\,2431,1243,3124,4312\,\\}$.

As transitivity turns a DAG into a poset, we now introduce the definition of
toric transitivity for a DAG, which will turn the corresponding toric DAG into
a toric poset.

A DAG $\vec{D}$ with vertex set $[n]$ is toric transitive if the existence of
a directed path $i_{1}\to i_{2}\to\dots\to i_{k}$ and $i_{1}\to i_{k}$ implies
the existence of $i_{a}\to i_{b}$ in $\vec{D}$ for all $1\leq a<b\leq k$. A
toric DAG $[\vec{D}]$ is a toric poset if $\vec{D}^{\prime}$ is toric
transitive for some $\vec{D}^{\prime}\in[\vec{D}]$, or equivalently from
[AGRR20] Proposition 3.10, for all representatives $\vec{D}^{\prime}$. A toric
poset is a total cyclic order if one (or equivalently, all, according to
Proposition 4.1) representative is a total linear order. In this case, we
usually use the corresponding cyclic permutation $[w]$ to denote the total
cyclic order $[\vec{D}]$.

######  Remark 4.3.

In this paper, we adopt the definition of toric posets from [AGRR20], which is
not quite the same as it was originally defined in [DMR16], but they are
essentially equivalent by [DMR16], Theorem 1.4.

For two toric DAGs $[\vec{D}_{1}],[\vec{D}_{2}]$ on the same vertex set $[n]$,
we say $[\vec{D_{2}}]$ extends $[\vec{D}_{1}]$ if there exist
$\vec{D}_{i}^{\prime}\in[\vec{D}_{i}^{\prime}]$ for $i=1,2$ such that
$\vec{D}_{2}^{\prime}$ extends $\vec{D}_{1}^{\prime}$. If, furthermore,
$[\vec{D}_{2}]$ is a total cyclic order corresponding to the cyclic
permutation $[w]$, we also say that $[w]$ torically extends $[\vec{D}_{1}]$.
Let $\mathcal{L}^{\operatorname{tor}}([\vec{D}])$ denote the set of cyclic
permutations $[w]$ which torically extend $[\vec{D}]$.

######  Example 4.4.

In the Example 3.1, both $[\vec{D}_{1}]=[2431]$ and $[\vec{D}_{2}]=[2413]$
torically extend $[\vec{D}_{3}]$. Moreover, they are the only total cyclic
orders that torically extend $[\vec{D}_{3}]$, namely
$\mathcal{L}^{\operatorname{tor}}([\vec{D}_{3}])=\\{[2431],[2413]\\}$.

In fact, Figure 1 lists all representatives of $[\vec{D}_{3}]$, and it is
straightforward to check that every total linear order linearly extending some
DAG in $[\vec{D}_{3}]$ is in either $[2431]$ or $[2413]$.

$\vec{D}_{3}$$2$$4$$3$$1$

$\vec{D}_{3}^{\prime}$$2$$4$$3$$1$

$\vec{D}_{3}^{\prime\prime}$$2$$4$$3$$1$

$\vec{D}_{3}^{\prime\prime\prime}$$2$$4$$3$$1$

$\vec{D}_{3}^{\prime\prime\prime\prime}$$2$$4$$3$$1$

Figure 1: All representatives of $[\vec{D}_{3}]$

######  Definition 4.5 (Enriched toric $[\vec{D}]$-partition).

An enriched toric $[\vec{D}]$-partition is a function
$f:[n]\to\mathbb{P^{\prime}}$ which is an enriched
$\vec{D^{\prime}}$-partition for at least one DAG $\vec{D}^{\prime}$ in
$[\vec{D}]$. Let $\mathcal{E}^{\operatorname{tor}}([\vec{D}])$ denote the set
of all enriched toric $[\vec{D}]$-partitions.

If $[\vec{D}]=[w]$ is a total cyclic order, the set of enriched toric
$[w]$-partitions is the union of the set of enriched $w^{\prime}$-partitions
for all representatives $w^{\prime}$ of $[w]$:

$\mathcal{E}^{\operatorname{tor}}([w])=\bigcup_{w^{\prime}\in[w]}\mathcal{E}(w^{\prime}).$
(4.1)

As in the linear case, we have the following fundamental lemma for the
decomposition of enriched toric $[\vec{D}]$-partitions. The proof is analogous
to [AGRR20], Lemma 3.15.

######  Lemma 4.6 (Fundamental lemma of enriched toric
$[\vec{D}]$-partitions).

For a DAG $\vec{D}$, the set of all enriched toric $[\vec{D}]$-partitions is a
disjoint union of the set of enriched toric $[w]$-partitions of all toric
extensions $[w]$ of $[\vec{D}]$:

$\mathcal{E}^{\operatorname{tor}}([\vec{D}])=\bigsqcup_{[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])}\mathcal{E}^{tor}([w]).$

###### Proof.

By the definition of $\mathcal{E}^{\operatorname{tor}}([\vec{D}])$, one has

$\mathcal{E}^{\operatorname{tor}}([\vec{D}])=\bigcup_{\vec{D^{\prime}}\in[\vec{D}]}\mathcal{E}(\vec{D^{\prime}}).$

In particular when $[\vec{D}]=[w]$ is a total cyclic order, it follows from
the Proposition 4.1 that,

$\mathcal{E}^{\operatorname{tor}}([w])=\bigcup_{w^{\prime}\in[w]}\mathcal{E}({w^{\prime}}).$

Hence,

$\displaystyle\mathcal{E}^{\operatorname{tor}}([\vec{D}])$
$\displaystyle=\bigcup_{\vec{D^{\prime}}\in[\vec{D}]}\mathcal{E}(\vec{D^{\prime}})$
$\displaystyle\overset{(i)}{=}\bigcup_{\vec{D^{\prime}}\in[\vec{D}]}\bigcup_{w^{\prime}\in\mathcal{L}(\vec{D^{\prime}})}\mathcal{E}(w^{\prime})$
$\displaystyle\overset{(ii)}{=}\bigcup_{[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])}\bigcup_{{w^{\prime}}\in[w]}\mathcal{E}(w^{\prime})$
$\displaystyle=\bigcup_{[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])}\mathcal{E}^{\operatorname{tor}}([w]).$

To justify these steps, first note that equality $(i)$ follows from Lemma 3.4.

As for equality $(ii)$, it suffices to show that
$w^{\prime}\in\mathcal{L}(\vec{D^{\prime}})$ for some
$\vec{D^{\prime}}\in[\vec{D}]$ if and only if $w^{\prime}\in[w]$ for some
$[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])$. For the forward
direction, if $w^{\prime}\in\mathcal{L}(\vec{D^{\prime}})$ for some
$\vec{D^{\prime}}\in[\vec{D}]$, then $[w^{\prime}]=[w]$ and $[w^{\prime}]$
torically extends $[\vec{D^{\prime}}]=[\vec{D}]$. While for the reverse
implication, given
$w^{\prime}\in[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])$, pick
$\vec{D^{\prime\prime}}\in[\vec{D}]$ and $w^{\prime\prime}\in[w]$ with
$w^{\prime\prime}$ linearly extending $\vec{D^{\prime\prime}}$, then
$[{w^{\prime\prime}}]=[{w^{\prime}}]=[w]$. It follows that there exists a
sequence of flips which takes $w^{\prime\prime}$ to $w^{\prime}$. Now applying
the same sequence of flips on $\vec{D^{\prime\prime}}$ will result in some
$\vec{D^{\prime}}$. One then has
$\vec{D^{\prime}}\in[\vec{D^{\prime\prime}}]=[\vec{D}]$ and
$w^{\prime}\in\mathcal{L}(\vec{D^{\prime}})$ as desired.

The assertion of disjointness follows directly from the fact that every
function $f:[n]\to\mathbb{P}^{\prime}$ has a unique linear permutation
$w\in\mathcal{S}_{n}$ such that $f$ is also an enriched $w$-partition, hence
an enriched toric $[w]$-partition. Such a linear permutation $w$ can be
similarly constructed as in the proof of Lemma 3.4, so the details are
omitted. This completes the proof. ∎

For the sake of convenience, we always assume the label set $[n]$, however, it
is note that the definition of (toric) DAGs, (toric) posets can be extended to
any finite subsets of $\mathbb{P}$ as labels, with all consequent conclusions
continuing to hold.

##  5 Weight enumerators and algebra of cyclic peaks

In this section, we first review the weight enumerators for enriched
$P$-partitions from Stembridge [Ste97] in section 5.1. These enumerators will
span an algebra $\Pi$ referred as the algebra of peaks, which is a graded
subring of $\operatorname{QSym}$. We then discuss its cyclic analogue for
enriched toric $[\vec{D}]$-partitions in section 5.2. In this case,
enumerators will generate the algebra of cyclic peaks $\Lambda$, which is a
graded subring of $\operatorname{cQSym}$.

###  5.1 Weight enumerator for enriched $\vec{D}$-partitions

Suppose $\vec{D}$ is a DAG on $[n]$. Define the weight enumerator for enriched
$\vec{D}$-partitions by the formal series

$\Delta_{\vec{D}}:=\sum_{f\in\mathcal{E}(\vec{D})}\prod_{i\in[n]}x_{|f(i)|},$

where $\mathcal{E}(\vec{D})$ is the set of enriched $\vec{D}$-partitions. By
the Fundamental Lemma 3.4, one has

$\Delta_{\vec{D}}=\sum_{w\in\mathcal{L}(\vec{D})}\Delta_{w}.$ (5.1)

It is clear from equation (3.1) that $\Delta_{w}$ is a homogeneous quasi-
symmetric function. More generally, $\Delta_{\vec{D}}$ is a homogeneous quasi-
symmetric function in $\operatorname{QSym}$.

It also follows from equation (3.1) that the weight enumerator $\Delta_{w}$
depends on the descent set $\operatorname{Des}w$. A less obvious but important
observation that $\Delta_{w}$ depends only on the peak set
$\operatorname{Pk}w$ will follow directly from the next proposition, proved by
Stembridge in [Ste97].

######  Proposition 5.1 ([Ste97], Proposition 2.2).

As a quasi-symmetric function, $\Delta_{w}$ has the following expansion of
monomial quasi-symmetric functions

$\Delta_{w}=\sum_{\begin{subarray}{c}E\subseteq[n-1]\colon\\\\[4.0pt]
\operatorname{Pk}w\subseteq E\cup(E+1)\end{subarray}}2^{|E|+1}M_{n,E},$ (5.2)

where the set $E+1$ is defined by (2.1).

######  Example 5.2.

From the Example 4.4, we have
$\mathcal{L}^{\operatorname{tor}}([\vec{D}_{3}])=\\{\,[2431],[2413]\,\\}$. By
the Definition 5.4 and equation (5.3) in next subsection, one has

$\displaystyle\Delta_{[\vec{D}_{3}]}^{\operatorname{cyc}}$
$\displaystyle=\Delta_{[2431]}^{\operatorname{cyc}}+\Delta_{[2413]}^{\operatorname{cyc}}$
$\displaystyle=\Delta_{2431}+\Delta_{4312}+\Delta_{3124}+\Delta_{1243}+\Delta_{2413}+\Delta_{4132}+\Delta_{1324}+\Delta_{3241}.$

Applying the proposition above, we can calculate each summand as follows:

$\displaystyle\Delta_{2431}$
$\displaystyle=\sum_{E\subseteq[3]\colon\\{2\\}\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{1\\}}+M_{4,\\{2\\}}),$
$\displaystyle\Delta_{4312}$
$\displaystyle=\sum_{E\subseteq[3]\colon\emptyset\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{1\\}}+M_{4,\\{2\\}}+M_{4,\\{3\\}})+2M_{4,\emptyset},$
$\displaystyle\Delta_{3124}$
$\displaystyle=\sum_{E\subseteq[3]\colon\emptyset\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{1\\}}+M_{4,\\{2\\}}+M_{4,\\{3\\}})+2M_{4,\emptyset},$
$\displaystyle\Delta_{1243}$
$\displaystyle=\sum_{E\subseteq[3]\colon\\{3\\}\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{2\\}}+M_{4,\\{3\\}}),$
$\displaystyle\Delta_{2413}$
$\displaystyle=\sum_{E\subseteq[3]\colon\\{2\\}\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{1\\}}+M_{4,\\{2\\}}),$
$\displaystyle\Delta_{4132}$
$\displaystyle=\sum_{E\subseteq[3]\colon\\{3\\}\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{2\\}}+M_{4,\\{3\\}}),$
$\displaystyle\Delta_{1324}$
$\displaystyle=\sum_{E\subseteq[3]\colon\\{2\\}\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{1\\}}+M_{4,\\{2\\}}),$
$\displaystyle\Delta_{3241}$
$\displaystyle=\sum_{E\subseteq[3]\colon\\{3\\}\subseteq
E\cup(E+1)}2^{|E|+1}M_{4,E}$
$\displaystyle=2^{4}M_{4,\\{1,2,3\\}}+2^{3}(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}})+2^{2}(M_{4,\\{2\\}}+M_{4,\\{3\\}}).$

Therefore, we have

$\begin{array}[]{rl}\Delta_{[\vec{D}_{3}]}^{\operatorname{cyc}}=&2^{4}\cdot
8M_{4,\\{1,2,3\\}}+2^{3}\cdot
8\left(M_{4,\\{1,2\\}}+M_{4,\\{1,3\\}}+M_{4,\\{2,3\\}}\right)+2^{2}\cdot
5\left(M_{4,\\{1\\}}+M_{4,\\{3\\}}\right)\\\\[4.30554pt] &+2^{2}\cdot
8M_{4,\\{2\\}}+2\cdot 2M_{4,\emptyset}\\\\[8.61108pt] =&2\cdot
2^{4}M^{cyc}_{(1,1,1,1)}+8\cdot 2^{3}M^{cyc}_{(2,1,1)}+5\cdot
2^{2}M^{cyc}_{(3,1)}+4\cdot 2^{2}M^{cyc}_{(2,2)}+2\cdot 2M^{cyc}_{(4)},\\\
\end{array}$

where the second equality follows from the Table 1.

As a counterpart, the weight enumerator $\Delta_{w}$ also has an expansion in
terms of another basis: the fundamental quasi-symmetric functions.

######  Proposition 5.3 ([Ste97], Proposition 3.5).

As a quasi-symmetric function, $\Delta_{w}$ has the following expansion of
fundamental quasi-symmetric functions

$\Delta_{w}=2^{\operatorname{pk}w+1}\sum_{D\subseteq[n-1]\colon\operatorname{Pk}w\subseteq
D\triangle(D+1)}F_{n,D}.$

Here $\triangle$ denotes the symmetric difference, that is, $D\triangle
E=(D\cup E)\setminus(D\cap E)$.

We say a subset $S\subseteq[n]$ is a peak set in $[n]$ if
$\operatorname{Pk}w=S$ for some $w\in\mathcal{S}_{n}$. For any peak set $S$ in
$[n]$, from the above proposition we can define an associated quasi-symmetric
function by

$K_{S}:=\Delta_{w},$

for any permutation $w$ with peak set $S$. It follows that
$\Delta_{w}=K_{\operatorname{Pk}w}$ and one can rewrite equation (5.1) as

$\Delta_{\vec{D}}=\sum_{w\in\mathcal{L}(\vec{D})}K_{\operatorname{Pk}w}.$

Let $\Pi_{n}$ denote the space of quasi-symmetric functions spanned by
$K_{S}$, taken over all peak sets in $[n]$ and set $\Pi=\oplus_{n\geq
0}\Pi_{n}$. In [Ste97] Stembridge referred to $\Pi$ as the algebra of peaks,
and proved that $\Pi$ is a graded subring of $\operatorname{QSym}$.

###  5.2 Weight enumerator for enriched toric $[\vec{D}]$-partitions

######  Definition 5.4.

For a given toric poset $[\vec{D}]$ with vertex set $[n]$, we define the
weight enumerator for enriched toric $[\vec{D}]$-partitions by the formal
power series

$\Delta_{[\vec{D}]}^{\operatorname{cyc}}:=\sum_{f\in\mathcal{E}^{\operatorname{tor}}([\vec{D}])}\prod_{i\in[n]}x_{|f(i)|}.$

Namely, for integer $k>0$ we assign the weight $x_{k}$ to both $f$-values $k$
and $-k$.

As a direct consequence of the Fundamental Lemma 4.6, one has

$\Delta_{[\vec{D}]}^{\operatorname{cyc}}=\sum_{[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])}\Delta_{[w]}^{\operatorname{cyc}}.$
(5.3)

Therefore, it suffices to discuss $\Delta_{[w]}^{\operatorname{cyc}}$ for
cyclic permutations $[w]$. It follows from the formula (4.1) that
$\Delta_{[w]}^{\operatorname{cyc}}$ can be expressed in terms of the weight
enumerators $\\{\Delta_{v}\\}$ as

$\Delta_{[w]}^{\operatorname{cyc}}=\sum_{v\in[w]}\Delta_{v}.$ (5.4)

Moreover, $\Delta_{[w]}^{\operatorname{cyc}}$ has the following expression.

######  Proposition 5.5.

For any given cyclic permutation $[w]$ of length $n$, we have

$\Delta_{[w]}^{cyc}=\sum_{\begin{subarray}{c}E\subseteq[n]:\\\\[4.0pt]
\operatorname{cPk}(w)\subseteq E\cup(E+1)\end{subarray}}2^{|E|}M_{n,E}^{cyc},$
(5.5)

with set $E+1$ defined by (2.1). The sum is independent on the choice of
representative $w$ of $[w]$.

###### Proof.

The independence of representatives is a result of the following two
observations:

  1. (a)

If $E$ and $E^{\prime}$ only differ by a cyclic shift, one has
$|E|=|E^{\prime}|$ and $M_{n,E}^{cyc}=M_{n,E^{\prime}}^{cyc}$.

  2. (b)

For two representatives $w$ and $w^{\prime}$ of $[w]$,

$\\{E\subseteq[n]:\operatorname{cPk}(w)\subseteq
E\cup(E+1)\\}=\\{E^{\prime}\subseteq[n]:\operatorname{cPk}(w^{\prime})\subseteq
E^{\prime}\cup(E^{\prime}+1)\\}+i,$

for some $i\in[n]$, namely, the two sets only differ by a cyclic shift.

Now fix a representative $w$ of $[w]$. We rewrite both sides of equation (5.5)
as follows:

$\mathrm{RHS}\overset{(i)}{=}\sum_{\begin{subarray}{c}F\subseteq[n]:\\\\[4.0pt]
\operatorname{cPk}(w)\subseteq F\cup(F+1)\end{subarray}}2^{|F|}\sum_{f\in
F}M_{n,(F-f)\cap[n-1]}\overset{(ii)}{=}\sum_{E\subseteq[n-1]}2^{|E|+1}\alpha_{E}M_{n,E}$
(5.6)

where $\alpha_{E}=\\#A_{E}$, with

$A_{E}=\\{(F,f):f\in F\subseteq[n]\text{ with }\operatorname{cPk}(w)\subseteq
F\cup(F+1),\;E=(F-f)\cap[n-1]\\}.$

Here equality $(i)$ is a result of applying equation (2.5) to transform from
cyclic monomial to monomial, while equality $(ii)$ is obtained by picking a
$M_{n,E}$ and calculating the coefficient. It is noted that in equality
$(ii)$, for each qualified pair $(F,f)\in A_{E}$, since $f\in F$, we have
$n\in F-f$. As a result, $E=(F-f)\cap[n-1]=(F-f)\setminus\\{n\\}$, or
equivalently, $F=(E+f)\cup\\{f\\}$. Therefore their cardinalities satisfy
$|F|=|E|+1$.

$\mathrm{LHS}\overset{(i)^{\prime}}{=}\sum_{v\in[w]}\Delta_{v}\overset{(ii)^{\prime}}{=}\sum_{v\in[w]}\sum_{\begin{subarray}{c}E\subseteq[n-1]:\\\\[4.0pt]
\operatorname{Pk}(v)\subseteq
E\cup(E+1)\end{subarray}}2^{|E|+1}M_{n,E}\overset{(iii)^{\prime}}{=}\sum_{E\subseteq[n-1]}\beta_{E}2^{|E|+1}M_{n,E}$
(5.7)

where $\beta_{E}=\\#B_{E}$ if we set

$B_{E}=\\{i\in[n]:(\operatorname{cPk}w-i)\cap[2,n-1]\subseteq E\cup(E+1)\\}.$

Equality $(i)^{\prime}$ follows from equation (5.4). Equality $(ii)^{\prime}$
is obtained by applying equation (5.2) and expressing the weight enumerators
$\Delta_{\pi}$ in terms of monomial quasi-symmetric functions.

As for equality $(iii)^{\prime}$, notice that by interchanging the order of
double summation, we have

$\sum_{v\in[w]}\sum_{\begin{subarray}{c}E\subseteq[n-1]:\\\\[4.0pt]
\operatorname{Pk}(v)\subseteq
E\cup(E+1)\end{subarray}}2^{|E|+1}M_{n,E}=\sum_{E\subseteq[n-1]}\gamma_{E}2^{|E|+1}M_{n,E}$

where $\gamma_{E}=\\#C_{E}$ if we set

$C_{E}=\\{v\in[w]:\operatorname{Pk}v\subseteq E\cup(E+1)\\}.$

From the observation

$\\{\\{\,\operatorname{Pk}(v):v\in[w]\,\\}\\}=\\{\\{\,(\operatorname{cPk}w-i)\cap[2,n-1]:i\in[n]\,\\}\\}$

we obtain $\\#C_{E}=\\#B_{E}$ and hence $\gamma_{E}=\beta_{E}$. This proves
that the equality $(iii)^{\prime}$ holds.

By comparing equations (5.6) and (5.7), it suffices to prove
$\alpha_{E}=\beta_{E}$ for every $E\subseteq[n-1]$, or equivalently, to
construct a bijection between $A_{E}$ and $B_{E}$.

Set $\theta_{E}:A_{E}\to B_{E}$ as $\theta_{E}(F,f)=f$. To prove that this map
is well-defined, it suffices to show that for each $(F,f)\in A_{E}$, we have
$f\in B_{E}$. It follows from the definition of $A_{E}$ that
$F-f=E\cup\\{n\\}$. Apply the operation on both sides of the equation
$\operatorname{cPk}(w)\subseteq F\cup(F+1)$, we get

$\operatorname{cPk}(w)-f\subseteq(F-f)\cup(F-f+1)=E\cup(E+1)\cup\\{1,n\\}.$

Hence $(\operatorname{cPk}w-f)\cap[2,n-1]\subseteq E\cup(E+1)$ and $f\in
B_{E}$.

Conversely, define $\sigma_{E}:B_{E}\to A_{E}$ by
$\sigma_{E}(f)=(\,(E+f)\cup\\{f\\},f\,)$. One can similarly check that this
map is well-defined by the strategy of verifying $\theta_{E}$.

It is straightforward to verify that $\theta_{E}$ and $\sigma_{E}$ are
inverse, hence we get a bijection between $A_{E}$ and $B_{E}$. This finishes
the proof. ∎

######  Remark 5.6.

As a direct corollary of the above proposition,
$\Delta_{[\vec{D}]}^{\operatorname{cyc}}$ is a homogeneous cyclic quasi-
symmetric function of degree $|w|$, and that
$\Delta_{[w]}^{\operatorname{cyc}}$ depends only on $\operatorname{cPk}[w]$,
or equivalently from Remark 1.2, $\operatorname{cPk}w$ for any representative
$w$.

######  Example 5.7.

Let us compute an example to examine the validity of the previous proposition.
In particular, suppose $w=1243$, $[w]=\\{1243,3124,4312,2431\\}$ and
$\operatorname{cPk}(w)=\\{3\\}$; $\pi=1324$, $[\pi]=\\{1324,4132,2413,3241\\}$
and $\operatorname{cPk}(\pi)=\\{2,4\\}$.

If we were to use the Proposition 5.5 for calculation, we first need all
possible choices of $E\subseteq[4]$ satisfying
$\\{3\\}=\operatorname{cPk}w\subseteq E\cup(E+1)$, which are

$\\{1,2,3,4\\},\\{1,2,3\\},\\{1,2,4\\},\\{1,3,4\\},\\{2,3,4\\},\\{1,2\\},\\{1,3\\},\\{2,3\\},\\{2,4\\},\\{3,4\\},\\{2\\},\\{3\\},$

with corresponding monomial cyclic quasi-symmetric functions

$\begin{array}[]{ccl}M^{cyc}_{(1,1,1,1)}&=&M^{cyc}_{4,\\{1,2,3,4\\}},\\\\[4.30554pt]
M^{cyc}_{(2,1,1)}&=&M^{cyc}_{4,\\{1,2,3\\}}=M^{cyc}_{4,\\{1,2,4\\}}=M^{cyc}_{4,\\{1,3,4\\}}=M^{cyc}_{4,\\{2,3,4\\}},\\\\[4.30554pt]
M^{cyc}_{(3,1)}&=&M^{cyc}_{4,\\{1,2\\}}=M^{cyc}_{4,\\{2,3\\}}=M^{cyc}_{4,\\{3,4\\}},\\\\[4.30554pt]
M^{cyc}_{(2,2)}&=&M^{cyc}_{4,\\{1,3\\}}=M^{cyc}_{4,\\{2,4\\}},\\\\[4.30554pt]
M^{cyc}_{(4)}&=&M^{cyc}_{4,\\{2\\}}=M^{cyc}_{4,\\{3\\}}.\\\ \end{array}$

Applying equation (5.5), we have

$\Delta_{[w]}^{cyc}=2^{4}M^{cyc}_{(1,1,1,1)}+4\cdot
2^{3}M^{cyc}_{(2,1,1)}+3\cdot 2^{2}M^{cyc}_{(3,1)}+2\cdot
2^{2}M^{cyc}_{(2,2)}+2\cdot 2M^{cyc}_{(4)}.$

Similarly for the consideration of $\pi$, all possible choices of
$E\subseteq[4]$ satisfying $\\{2,4\\}=\operatorname{cPk}\pi\subseteq
E\cup(E+1)$ are as follows:

$\\{1,2,3,4\\},\\{1,2,3\\},\\{1,2,4\\},\\{1,3,4\\},\\{2,3,4\\},\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\},$

hence by equation (5.5),

$\Delta_{[\pi]}^{cyc}=2^{4}M^{cyc}_{(1,1,1,1)}+4\cdot
2^{3}M^{cyc}_{(2,1,1)}+2\cdot 2^{2}M^{cyc}_{(3,1)}+2\cdot
2^{2}M^{cyc}_{(2,2)}.$

It follows from the calculation above that

$\Delta_{[w]}^{cyc}+\Delta_{[\pi]}^{cyc}=2\cdot
2^{4}M^{cyc}_{(1,1,1,1)}+8\cdot 2^{3}M^{cyc}_{(2,1,1)}+5\cdot
2^{2}M^{cyc}_{(3,1)}+4\cdot 2^{2}M^{cyc}_{(2,2)}+2\cdot 2M^{cyc}_{(4)},$

which is exactly the same enumeration as what we obtained in the Example 5.2.
This example verifies the validity of previous proposition.

###  5.3 Algebra of cyclic peaks

We say $S$ is a cyclic peak set in $[n]$ if there exists some
$w\in\mathcal{S}_{n}$ with $\operatorname{cPk}w=S$. It follows from
Proposition 5.5 that, for any cyclic peak set $S$ in $[n]$, we can define an
associated cyclic quasi-symmetric function by

$K^{\operatorname{cyc}}_{S}:=\Delta^{\operatorname{cyc}}_{[w]},$

for any permutation $w$ with $\operatorname{cPk}w=S$. One can observe
$\Delta^{\operatorname{cyc}}_{[w]}=K^{\operatorname{cyc}}_{\operatorname{cPk}w}$
and rewrite equation (5.3) as

$\Delta^{\operatorname{cyc}}_{[\vec{D}]}=\sum_{[w]\in\mathcal{L}^{\operatorname{tor}}([\vec{D}])}K^{\operatorname{cyc}}_{\operatorname{cPk}w}.$

Moreover, it follows from the formula (4.1) that $K^{\operatorname{cyc}}_{S}$
can be expressed in terms of quasi-symmetric functions $\\{K_{T}\\}$ as

$K^{\operatorname{cyc}}_{S}=\sum_{\sigma\in[w]}K_{\operatorname{Pk}\sigma},$

where $\operatorname{cPk}w=S$.

Let $\Lambda_{n}$ denote the space of cyclic quasi-symmetric functions spanned
by $K^{\operatorname{cyc}}_{S}$ where $S$ ranges over cyclic peak sets in
$[n]$, and set $\Lambda=\oplus_{n\geq 0}\Lambda_{n}$. We now will show that
$\Lambda$ is an algebra by proving that the product of
$K^{\operatorname{cyc}}_{U}$ and $K^{\operatorname{cyc}}_{T}$ is a linear
combination of $K^{\operatorname{cyc}}_{S}$’s, and we call $\Lambda$ the
algebra of cyclic peaks.

######  Lemma 5.8.

The $K^{\operatorname{cyc}}_{S}$ are linearly independent, where $S$ are
distinct cyclic peak sets up to cyclic shift.

###### Proof.

In this proof, we totally order the subsets of $[n-1]$, first by cardinality,
then by the lexicographic order. We therefore have

$\emptyset\lhd\\{1\\}\lhd\\{2\\}\lhd\cdots\lhd\\{n-1\\}\lhd\\{1,2\\}\lhd\\{1,3\\}\lhd\cdots\lhd\\{1,n-1\\}\lhd\\{2,3\\}\lhd\\{2,4\\}\lhd\cdots.$

One can similar order the compositions of $n$ as

$(n)\lhd(1,n-1)\lhd(2,n-2)\lhd\cdots\lhd(1,1,n-2)\lhd(1,2,n-3)\lhd\cdots\lhd(1,n-2,1)\lhd\cdots.$

Notice that $K^{\operatorname{cyc}}_{S}=K^{\operatorname{cyc}}_{S^{\prime}}$
if two sets $S$ and $S^{\prime}$ only differ by a cyclic shift, we will always
assume the index $S$ to be the least among all its cyclic shifts. It is not
hard to see that $\psi(S)$ is also the least composition among all its cyclic
shifts, where $\psi$ is defined by equation (2.3).

We now show that the matrix of $\\{K^{\operatorname{cyc}}_{S}\\}$ with respect
to $\\{M^{\operatorname{cyc}}_{n,L}\\}$ is a matrix of full rank. Then the
linear independence of $\\{K^{\operatorname{cyc}}_{S}\\}$ will follow
immediately from the fact that the monomial cyclic quasi-symmetric functions
form a basis of $\operatorname{cQSym}$.

Let us fix $n$ and suppose $\\{S_{1}\lhd S_{2}\lhd\cdots\lhd S_{m}\\}$ is the
set of all distinct cyclic peak sets in $[n]$. Given a
$K^{\operatorname{cyc}}_{S}$, suppose $|S|=k$ and
$S=\\{1=s_{1}<s_{2}<\cdots<s_{k}\\}$. Here all indices are taken modulo $k$
unless otherwise noted. To each $K_{S}^{\operatorname{cyc}}$ we associate a
corresponding monomial quasi-symmetric function by
$F(K^{\operatorname{cyc}}_{S})=M^{\operatorname{cyc}}_{n,f(S)}$, where

$f(S)=\\{s_{1},s_{2}-1,\cdots,s_{k}-1\\}.$

Since $S$ is a cyclic peak set, elements in $f(S)$ are distinct. So $f(S)$ is
a set and $F$ is well-defined. If one denotes $\psi(S)$ by
$(\alpha_{1},\alpha_{2},\dots,\alpha_{k})$, then

$\psi(f(S))=(\alpha_{1}-1,\alpha_{2},\dots,\alpha_{k-1},\alpha_{k}+1).$

Also notice that the assumption of $S$ being the least ensures that $f(S)$ is
also the least among all its cyclic shifts. It follows that $F$ is injective.

Claim: The matrix of $\\{K^{\operatorname{cyc}}_{S}\\}$ related to
corresponding $\\{M^{\operatorname{cyc}}_{n,f(S)}\\}$ is invertible. Moreover,
this matrix is an upper triangular matrix with nonzero diagonal entries.

Let $A_{i,j}$ denote the coefficient of $M^{\operatorname{cyc}}_{n,f(S_{j})}$
in the expression of $K^{\operatorname{cyc}}_{S_{i}}$ and set
$A=\\{A_{i,j}\\}$ to be the $m\times m$ matrix that we need to consider. To
prove that $A$ is an upper triangular matrix, it suffices to show that
$A_{i,j}=0$ if $i>j$, or equivalently, the term
$M^{\operatorname{cyc}}_{n,f(S_{j})}$ does not appear in the expression of
$K^{\operatorname{cyc}}_{S_{i}}$ if $S_{j}\lhd S_{i}$.

Suppose towards a contradiction that $A_{i,j}\neq 0$ for some $i>j$. It then
follows from the Proposition 5.5 that there exists some $E\subseteq[n]$ such
that

$S_{i}\subseteq E\cup(E+1)\quad\text{and}\quad
M^{\operatorname{cyc}}_{n,f(S_{j})}=M^{\operatorname{cyc}}_{n,E}.$

Since $S_{i}$ is a cyclic peak set, $e$ and $e+1$ cannot be in $S_{i}$ at the
same time. It then follows from the assumption $S_{i}\subseteq E\cup(E+1)$
that $|E|\geq|S_{i}|$. The other condition $i>j$ implies $S_{i}\rhd S_{j}$,
hence $|S_{i}|\geq|S_{j}|=|f(S_{j})|=|E|$. Therefore, we only need to consider
the cases when $S_{i}$ and $S_{j}$ have the same cardinality.

Suppose

$\psi(S_{i})=(\alpha_{1},\alpha_{2},\dots,\alpha_{k}),\quad\psi(S_{j})=(\beta_{1},\beta_{2},\dots,\beta_{k}),\quad\psi(E)=(\beta_{1}^{\prime},\beta_{2}^{\prime},\dots,\beta_{k}^{\prime}).$

Then
$\psi(E)=(\beta_{1}^{\prime},\beta_{2}^{\prime},\dots,\beta_{k}^{\prime})$ is
a cyclic shift of
$\psi(f(S_{j}))=(\beta_{1}-1,\beta_{2},\dots,\beta_{k-1},\beta_{k}+1)$. Assume

$(\beta_{q}^{\prime},\beta_{q+1}^{\prime},\dots,\beta_{q-1}^{\prime})=(\beta_{1}-1,\beta_{2},\dots,\beta_{k-1},\beta_{k}+1),$

for some $q\in[k]$. Notice from the condition $S_{i}\subseteq E\cup(E+1)$, the
cardinality relation $|S_{i}|=|E|$, and the fact that $S_{i}$ does not have
consecutive elements as a cyclic peak set, each $a\in S_{i}$ corresponds to a
unique $a^{\prime}\in E$ where $a^{\prime}$ equals $a$ or $a-1$ (considered
modulo $n$). A crucial observation therefore follows:

Suppose $a,b\in S_{i}$ correspond to $a^{\prime},b^{\prime}\in E$
respectively, then $|(a-b)-(a^{\prime}-b^{\prime})|\leq 1$.

Equivalently, one has, for $r,s\in[k]$,

$|(\alpha_{r}+\cdots+\alpha_{s})-(\beta_{r}^{\prime}+\cdots+\beta_{s}^{\prime})|\leq
1.$ (5.8)

In particular,

$\alpha_{1}\leq\alpha_{q}\leq\beta_{q}^{\prime}+1=(\beta_{1}-1)+1=\beta_{1},$

where the first inequality comes from the fact that $S_{i}$ is the least among
its cyclic shifts, and the second one follows from (5.8) by taking $r=s=q$.
Note that $S_{j}\lhd S_{i}$ implies $\psi(S_{j})\lhd\psi(S_{i})$, so
$\beta_{1}\leq\alpha_{1}$. Thus, it must be

$\alpha_{1}=\alpha_{q}=\beta_{q}^{\prime}+1=\beta_{1}.$

It then follows from the inequality (5.8) that for any $r\in[k-1]$,

$\alpha_{q}+\cdots+\alpha_{q+r}\leq\beta_{q}^{\prime}+\cdots+\beta_{q+r}^{\prime}+1=\beta_{1}+\cdots+\beta_{1+r}+\delta_{1+r,k},$

where $\delta_{1+r,k}=1$ if $1+r=k$ and $0$ otherwise.

If $\alpha_{q+r}=\beta_{1+r}$ for every $r\in[k-2]$, then $\psi(S_{i})$ is a
cyclic shift of $\psi(S_{j})$, which contradicts to our assumption $S_{i}\rhd
S_{j}$. Now assume $t\in[k-2]$ is the smallest index such that
$\alpha_{q+t}<\beta_{1+t}$, then

$(\alpha_{q},\dots,\alpha_{q+t})\lhd(\beta_{1},\dots,\beta_{1+t}).$

Combined with the inequality
$(\alpha_{1},\dots,\alpha_{1+t})\unlhd(\alpha_{q},\dots,\alpha_{q+t})$, as
$(\alpha_{1},\dots,\alpha_{k})$ is the least among all its cyclic shifts, one
has

$(\alpha_{1},\dots,\alpha_{1+t})\lhd(\beta_{1},\dots,\beta_{1+t}).$

This implies that $\psi(S_{i})\lhd\psi(S_{j})$, which is a contradiction to
the assumption $S_{i}\rhd S_{j}$. Moreover, it is not hard to see that
$A_{i,i}\neq 0$ as $S_{i}\subseteq f(S_{i})\cup(f(S_{i})+1)$, therefore the
diagonal entries are nonzero. This proves the claim, hence the lemma. ∎

Define the union of digraphs $\vec{D}\biguplus\vec{E}$ to be the digraph with
vertices $V(\vec{D})\cup V(\vec{E})$ and arcs $A(\vec{D})\cup A(\vec{E})$.
Next result follows easily from the definition.

######  Proposition 5.9.

Suppose $\vec{D}$ and $\vec{E}$ are two DAGs on disjoint subsets of
$\mathbb{P}$, then

$\Delta^{\operatorname{cyc}}_{[\vec{D}\biguplus\vec{E}]}=\Delta^{\operatorname{cyc}}_{[\vec{D}]}\cdot\Delta^{\operatorname{cyc}}_{[\vec{E}]}.$
(5.9)

The proposition above provides us a proof of $\Lambda$ being an algebra from
the combinatorial aspect.

######  Proposition 5.10.

$\Lambda$ is a graded subring of $\operatorname{cQSym}$.

###### Proof.

As a subspace of $\operatorname{cQSym}$, $\Lambda$ naturally inherits the
addition and multiplication operations from $\operatorname{cQSym}$.

To show that $\Lambda$ is closed under multiplication, take
$K^{\operatorname{cyc}}_{U}\in\Lambda_{m}$ and
$K^{\operatorname{cyc}}_{T}\in\Lambda_{n}$, where $U$ and $T$ are cyclic peak
sets in $[m]$ and $[n]$ respectively, namely, there exist
$\pi\in\mathcal{S}_{m}$ and $w\in\mathcal{S}_{n}$ such that
$\operatorname{cPk}\pi=U,\;\operatorname{cPk}w=T$. For the purpose of
constructing two corresponding disjoint DAGs, we standardize
$w=w_{1}w_{2}\ldots w_{n}\in\mathcal{S}_{n}$ to $\\{m+1,m+2,\ldots,m+n\\}$,
that is, construct $w^{\prime}=w^{\prime}_{1}w_{2}^{\prime}\ldots
w_{n}^{\prime}$ where $w_{i}^{\prime}=w_{i}+m$ for $i\in[n]$. As a consequence
of equation (5.9), we have

$K^{\operatorname{cyc}}_{U}\cdot
K^{\operatorname{cyc}}_{T}=\Delta^{\operatorname{cyc}}_{[\pi]}\cdot\Delta^{\operatorname{cyc}}_{[w^{\prime}]}=\Delta^{\operatorname{cyc}}_{[\pi]\biguplus[w^{\prime}]}=\sum_{\sigma\in\mathcal{L}^{\operatorname{tor}}([\pi]\biguplus[w^{\prime}])}K^{\operatorname{cyc}}_{\operatorname{cPk}\sigma}.$
(5.10)

This completes the proof. ∎

In terms of another basis of $\operatorname{cQSym}$, the fundamental cyclic
quasi-symmetric functions, $K^{\operatorname{cyc}}_{S}$ has the following
expansion.

######  Proposition 5.11.

For any cyclic peak set $S$ in $[n]$, we have

$2^{-|S|}K^{\operatorname{cyc}}_{S}=\sum_{\begin{subarray}{c}E\subseteq[n]:\\\\[2.0pt]
S\subseteq E\triangle(E+1)\end{subarray}}F^{cyc}_{n,E},$

where $\triangle$ denotes symmetric difference.

###### Proof.

Since $F^{cyc}_{n,E}=\sum_{F\supseteq E}M^{cyc}_{n,F}$, the coefficient of
$M^{cyc}_{n,F}$ on the right side of the above expansion is $|\\{E\subseteq
F:S\subseteq E\triangle(E+1)\\}|$.

To count this set, we need the following observations for possible choices of
$E$. For each $k\in S\subseteq E\triangle(E+1)$, exactly one of $k-1$ and $k$
is in $E$, since if both are in $E$ then $k$ is not in the symmetric
difference. It follows from $E\subseteq F$ that at least one of $k-1$ and $k$
is in $F$. So one has the following two cases:

  1. (1)

If both $k,k-1\in F$, $E$ must contain exactly one of $k$ or $k-1$. Otherwise,
suppose $E$ contains both $k$ and $k-1$, then $k\notin E\triangle(E+1)$ by the
definition of symmetric difference. This is a contradiction to the assumption
$k\in S\subseteq E\triangle(E+1)$.

  2. (2)

If only one of $k,k-1\in F$, then $E$ must contain whichever $F$ does.

It is noted that the restriction above only involve two adjacent numbers.

Also notice that if $k\in F$ but neither $k$ nor $k+1$ is in $S$, then $k$ is
free to be in $E$ or not. Denote

$\displaystyle S_{1}$ $\displaystyle=\\{k\in S\mid\text{both $k,k-1$ are in
$F$}\\},$ $\displaystyle S_{2}$ $\displaystyle=\\{k\in S\mid k\in E,k-1\notin
F\\},$ $\displaystyle S_{3}$ $\displaystyle=\\{k\in S\mid k\notin E,k-1\in
F\\}.$

Then we have a set partition $S=S_{1}\cup S_{2}\cup S_{3}$. Therefore if we
denote $s_{i}=\\#S_{i}$ for $i\in\\{1,2,3\\}$, we have
$|S|=s_{1}+s_{2}+s_{3}$.

Moreover, if we denote

$T=\\{k\in F\mid\text{none of $k,k+1$ is in $S$}\\},$

notice from the definition of peak set $S$, numbers in $S$ are not adjacent.
Hence we have the following set partition of $F$:

$F=S_{1}\cup(S_{1}-1)\cup S_{2}\cup(S_{3}-1)\cup T.$

It follows that $|F|=2s_{1}+s_{2}+s_{3}+t$ with $t=|T|$. Hence, the number of
choices for $E$ is

$2^{s_{1}+t}=2^{s_{1}+|F|-2s_{1}-s_{2}-s_{3}}=2^{|F|-(s_{1}+s_{2}+s_{3})}=2^{|F|-|S|}.$

So we have

$\sum\limits_{\begin{subarray}{c}E\subseteq[n]:\\\\[2.0pt] S\subseteq
E\triangle(E+1)\end{subarray}}F^{cyc}_{n,E}=\sum\limits_{\begin{subarray}{c}F\subseteq[n]:\\\\[2.0pt]
S\subseteq F\cup(F+1)\end{subarray}}2^{|F|-|S|}M^{cyc}_{n,F}.$

By equation (5.5), this quantity is $2^{-|S|}K^{cyc}_{S}$. ∎

###  5.4 Order polynomials $\Omega^{\operatorname{cyc}}([\vec{D}],m)$

Given a DAG $\vec{D}$, we can define the order polynomial of enriched
$\vec{D}$-partitions, $\Omega(\vec{D},m)$, by

$\Omega(\vec{D},m)=\Delta_{\vec{D}}(1^{m}),$

where $\Delta_{\vec{D}}(1^{m})$ means that we set $x_{1}=\cdots=x_{m}=1$, and
$x_{k}=0$ for $k>m$. In fact, $\Omega(\vec{D},m)$ counts the number of
enriched $\vec{D}$-partitions with absolute value at most $m$.

Moreover, Stembridge computed the corresponding generating function as
follows:

######  Theorem 5.12 ([Ste97], Theorem 4.1).

For a given $w\in\mathcal{S}_{n}$, one has

$\sum_{m}\Omega(w,m)t^{m}=\frac{1}{2}\left(\frac{1+t}{1-t}\right)^{n+1}\left(\frac{4t}{(1+t)^{2}}\right)^{1+\operatorname{pk}w}.$
(5.11)

It is not hard to see that $\Omega(\vec{D},t)$ is indeed a polynomial in $t$.
Suppose $\vec{D}$ has vertex set $V$ and that $|V|=n$, then

$\Omega(\vec{D},m)=\sum_{k=1}^{n}c_{k}\binom{m}{k},$

where $c_{k}$ denotes the number of $f\in\mathcal{E}(\vec{D})$ such that
$\\{|f(x)|\colon x\in V\\}=[k]$. For any fixed $k$, $\binom{m}{k}$ is a
polynomial in $m$ of degree $k$. Since $c_{k}$ and $n$ are constants, it
follows that the summation is also a polynomial in $m$. This verifies that
$\Omega([\vec{D}],t)$ is a polynomial.

Similarly, we define the order polynomial of enriched toric
$[\vec{D}]$-partitions, $\Omega^{\operatorname{cyc}}([\vec{D}],m)$, by

$\Omega^{\operatorname{cyc}}([\vec{D}],m)=\Delta^{\operatorname{cyc}}_{[\vec{D}]}(1^{m}).$

The following result is the toric analogue of formula (5.11).

######  Proposition 5.13.

Given $w\in\mathcal{S}_{n}$, then

$\sum_{m}\Omega^{\operatorname{cyc}}([w],m)t^{m}=\left(\frac{4t}{(1+t)^{2}}\right)^{\operatorname{cpk}w}\left(\frac{1+t}{1-t}\right)^{n-1}\left(\operatorname{cpk}w+\frac{2nt}{(1-t)^{2}}\right).$
(5.12)

This right side of the equation does not depend on the choice of
representative $w$, as they all have the same cyclic peak number.

###### Proof.

By the definition of order polynomial,

$\displaystyle\sum_{m}\Omega^{\operatorname{cyc}}([w],m)t^{m}$
$\displaystyle=\sum_{m}\Delta^{\operatorname{cyc}}_{[w]}(1^{m})t^{m}$
$\displaystyle=\sum_{m}\sum_{v\in[w]}\Delta_{v}(1^{m})t^{m}$
$\displaystyle=\sum_{v\in[w]}\sum_{m}\Delta_{v}(1^{m})t^{m}$
$\displaystyle=\sum_{v\in[w]}\sum_{m}\Omega(v,m)t^{m}$
$\displaystyle=\sum_{v\in[w]}\frac{1}{2}\left(\frac{1+t}{1-t}\right)^{n+1}\left(\frac{4t}{(1+t)^{2}}\right)^{1+\operatorname{pk}v}$
$\displaystyle=\frac{1}{2}\left(\frac{1+t}{1-t}\right)^{n+1}\sum_{v\in[w]}\left(\frac{4t}{(1+t)^{2}}\right)^{1+\operatorname{pk}v},$

where the last equality is obtained by applying (5.11).

Observe that each representative of $[w]$ will start with a peak, end with a
peak or none of the two ends are peaks, which will result in the peak number
$\operatorname{cpk}w-1$, $\operatorname{cpk}w-1$ or $\operatorname{cpk}w$
respectively. And the number of those representatives with a peak element at
one end is $2\operatorname{cpk}w$. It follows from the discussion above that

$\displaystyle\sum_{m}\Omega^{\operatorname{cyc}}([w],m)t^{m}$
$\displaystyle=\frac{2\operatorname{cpk}w}{2}\left(\frac{1+t}{1-t}\right)^{n+1}\left(\frac{4t}{(1+t)^{2}}\right)^{\operatorname{cpk}w}+\frac{n-2\operatorname{cpk}w}{2}\left(\frac{1+t}{1-t}\right)^{n+1}\left(\frac{4t}{(1+t)^{2}}\right)^{1+\operatorname{cpk}w}$
$\displaystyle=\left(\frac{4t}{(1+t)^{2}}\right)^{\operatorname{cpk}w}\left(\frac{1+t}{1-t}\right)^{n-1}\left(\operatorname{cpk}w+\frac{2nt}{(1-t)^{2}}\right).$

This completes the proof. ∎

Notice that by taking the coefficient of $t^{m}$ on both sides of equation
(5.12), one can get the expression of the order polynomial of enriched
$\vec{D}$-partitions $\Omega^{\operatorname{cyc}}([w],m)$ in an algebraic
manner. It would be desirable to derive $\Omega^{\operatorname{cyc}}([w],m)$
combinatorially. For this purpose, we first give a combinatorial proof for the
order polynomial in the linear case. We will define the $(w,m)$-marking on the
permutation $w$ with a given positive integer $m$. Then we will show that each
$(w,m)$-marking corresponds to exactly $2^{2\operatorname{pk}+1}$ enriched
$w$-partitions with absolute value at most $m$.

Suppose $w\in\mathcal{S}_{n}$, $m\in\mathbb{P}$. One can naturally extend
$w=w_{1}\ldots w_{n}$ to $w^{\prime}=w_{0}w_{1}\ldots w_{n}w_{n+1}$ where
$w_{0}=w_{n+1}=\infty$. Let $R_{1}$ be the longest decreasing initial factor
of $w^{\prime}$. Now let $v$ denote $w^{\prime}$ with $R_{1}$ deleted and let
$R_{2}$ be the longest increasing initial factor of $w^{\prime}$. Continue in
this way and alternate decreasing and increasing factors, we can get a
factorization of $w^{\prime}$. We will call the factors runs and the
corresponding indices run indices. We denote by $I_{j}$ the set of run indices
corresponding to factor set $R_{j}$. It is noted that any extension
$w^{\prime}$ will start with a decreasing run and end with an increasing run,
which implies that the number of runs is always even.

Take

$w=\begin{array}[]{cccccc}1&2&3&4&5&6\\\ \textbf{1}&4&{\bf 3}&{\bf
2}&5&6\end{array}$

as an example. The corresponding natural extension

$w^{\prime}=\begin{array}[]{cccccccc}0&1&2&3&4&5&6&7\\\
\textbf{$\infty$}&\textbf{1}&4&{\bf 3}&{\bf 2}&5&6&\text{$\infty$}\end{array}$

has four runs

$R_{1}=\infty 1,\;R_{2}=4,\;R_{3}=32,\;R_{4}=56\infty,$

where the decreasing runs are in bold, and corresponding set of run indices
are

$I_{1}=\\{0,1\\},\,\;I_{2}=\\{2\\},\;I_{3}=\\{3,4\\},\;I_{4}=\\{5,6,7\\}.$

Suppose permutation $w^{\prime}$ has $r$ runs. We have the following
observations:

  1. 1.

The parity of $i$ indicates the trending of each run $R_{i}$. If $i$ is even,
$R_{i}$ is increasing. If $i$ is odd, then $R_{i}$ is decreasing.

  2. 2.

The total number of runs $r$ is closely related to the peak number
$\operatorname{pk}w$:

$r=2\operatorname{pk}w+2.$

We will decorate permutations with bars and marks. One can put bars between
adjacent columns in two-line notation (including the space before the first
column and the space after the last), whereas a column of $w$ with index
$i\in[n]$ can be marked if and only if $i,i+1\in I_{j}$ for some $j$. In other
word, $i$ and $i+1$ are in the same run index set. There can be multiple bars
between two adjacent columns and we count them with multiplicity, while each
column can be marked at most once. We will denote by ${\cal M}_{w}$ the set of
indices where the corresponding columns can be marked,

${\cal M}_{w}:=\\{i\in[n]\mid\text{ $i,i+1\in I_{j}$ for some $j$}\\}.$

We note that for a given $w$, the cardinality of the complement of set ${\cal
M}_{w}$ in $[n]$ is $2\operatorname{pk}w+1$. Equivalently, one has $|{\cal
M}_{w}|=n-2\operatorname{pk}w-1$.

######  Definition 5.14 ($(w,m)$-marking).

Suppose $w$ is a linear permutation, $m$ is a positive integer. A
$(w,m)$-marking is a marking of $w$ using $b$ bars and $d$ marked columns,
satisfying that $b+d=m-1-\operatorname{pk}w$.

######  Example 5.15.

If we set $w=143256$ as usual, and take $m=5$, then a $(w,5)$-marking has $b$
bars and $d$ marked columns, such that $b+d=3$. Two $(w,5)$-markings are
provided as follows. Both have two bars and one marked column, where the
marked column is in blue.

$\begin{array}[]{cc||cccc}1&2&3&4&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
5}&6\\\
1&4&3&2&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
5}&6\end{array}\qquad\qquad\begin{array}[]{cc|cccc|}1&2&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
3}&4&5&6\\\
1&4&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
3}&2&5&6\end{array}$

######  Proposition 5.16.

For a given $w\in\mathcal{S}_{n}$, one has

$\Omega(w,m)=2^{2\operatorname{pk}w+1}\sum_{k=0}^{m-1-\operatorname{pk}w}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{pk}w-1}{m-1-\operatorname{pk}w-k},$
(5.13)

where $(\binom{n+1}{k})$ denotes the number of multisets on $[n+1]$ with
cardinality $k$.

###### Proof.

It is clear by definition that the number of possible choices for
$(w,m)$-markings is

$\sum_{b+d=m-1-\operatorname{pk}w}\left(\binom{n+1}{b}\right)\binom{n-2\operatorname{pk}w-1}{d}=\sum_{k=0}^{m-1-\operatorname{pk}w}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{pk}w-1}{m-1-\operatorname{pk}w-k}.$

Therefore, it suffices to construct a $2^{2\operatorname{pk}w+1}$-to-one map
from the set of enriched $w$-partitions with absolute value at most $m$ to the
set of all $(w,m)$-markings.

Given an enriched $w$-partition $f$ with absolute value at most $m$, we can
inductively associate it with a unique $(w,m)$-marking as follows:

We first determine whether column $k$ gets marked for those $k\in{\cal
M}_{w}$. Suppose $k\in I_{i}$ for some $i\in[r]$. We mark column $k$ if and
only if $\delta_{k}+\gamma_{k}=1$ where

$\delta_{k}:=\delta\,(\text{$i$ is even and
$f(w_{k})<0$}),\qquad\gamma_{k}:=\delta\,(\text{$i$ is odd and
$f(w_{k})>0$}).$

Here the Kronecker function on a statement $R$ is defined by

$\delta(R)=\begin{cases}1&\text{if $R$ is true},\\\ 0&\text{if $R$ is
false}.\end{cases}.$

As for the placement of bars, we start by putting $|f(w_{1})|-1$ bars before
the first column. Inductively, suppose $k\in I_{i}$ for some $k\in[2,n]$,
$i\in[r]$ and that we have already constructed the markings and bars on and
before the $(k-1)$st column. Then the number of marks and bars strictly before
the $k$th column is constructed to be

$|f(w_{k})|-\lceil i/2\rceil-\delta_{k}.$ (5.14)

Finally we add bars after the last column so that the total number of marks
and bars is $m-\operatorname{pk}w-1$. In this manner, we can inductively
define a unique $(w,m)$-marking for $f$.

We must verify that the constructed marking is indeed a $(w,m)$-marking.
Firstly we will verify that the construction before the $n$th column is
possible in that (5.14) is a weakly increasing function of $k$ and strictly
increasing from the $k$th to the $(k+1)$st term if column $k$ is marked. In
other words, if $k\in I_{i}$ and $k+1\in I_{j}$, it suffices to show

$|f(w_{k})|-\lceil i/2\rceil-\delta_{k}\leq|f(w_{k+1})|-\lceil
j/2\rceil-\delta_{k+1},$

for all $k$, and that

$|f(w_{k})|-\lceil i/2\rceil-\delta_{k}<|f(w_{k+1})|-\lceil
j/2\rceil-\delta_{k+1},$

when the column $k$ is marked. Notice that

$(\delta_{k}+\gamma_{k})\cdot\delta(k\in{\cal M}_{w})=1$

if column $k$ is marked and $0$ otherwise. Hence it suffices to show that

$|f(w_{k})|-\lceil
i/2\rceil-\delta_{k}+(\delta_{k}+\gamma_{k})\cdot\delta(k\in{\cal
M}_{w})\leq|f(w_{k+1})|-\lceil j/2\rceil-\delta_{k+1}.$ (5.15)

By the definition of enriched $P$-partitions, we have

$|f(w_{k})|\leq|f(w_{k+1})|.$ (5.16)

Let us consider the following cases:

  1. (a)

If $j=i$, it follows that $k\in{\cal M}_{w}$. Hence the inequality (5.15)
simplifies to prove

$|f(w_{k})|+\gamma_{k}\leq|f(w_{k+1})|-\delta_{k+1}.$

If $i$ is even, then $\gamma_{k}=0$ and $w_{k}<w_{k+1}$. By the inequality
(5.16), one only needs to consider whether the inequality holds when
$\delta_{k+1}=1$, which implies $f(w_{k+1})<0$. It follows from the definition
of enriched $P$-partitions that $f(w_{k+1})\succeq f(w_{k})-1$, or
equivalently, $|f(w_{k+1})|\geq|f(w_{k})|+1$, which proves (5.15) in this
case. The proof is similar when $i$ is odd.

  2. (b)

If $j=i+1$, then $k\notin{\cal M}_{w}$. The inequality (5.15) reduces to

$|f(w_{k})|-\lceil
i/2\rceil-\delta_{k}\leq|f(w_{k+1})|-\lceil(i+1)/2\rceil-\delta_{k+1}.$

If $i$ is even, then $w_{k}<w_{k+1}$, $\lceil i/2\rceil+1=\lceil(i+1)/2\rceil$
and $j$ is odd, hence $\delta_{k+1}=0$. Therefore one only needs to prove

$|f(w_{k})|-\delta_{k}\leq|f(w_{k+1})|-1.$

The inequality clearly holds when $\delta_{k}=1$. So it suffices to consider
the case when $\delta_{k}=0$. In this case, $f(w_{k})>0$, hence
$f(w_{k+1})\succeq-(f(w_{k})+1)$ from the definition of enriched
$P$-partitions, or equivalently $|f(w_{k+1})|\geq|f(w_{k})|+1$, which proves
(5.15). The case when $i$ is odd is similar and left to the readers.

Secondly, one also needs to check that it is possible to add bars (possibly
$0$) after the last column so that the total number of marks and bars is
$m-\operatorname{pk}w-1$. In other words, the number of bars we add after the
$n$th column is nonnegative. Notice that $\lceil\frac{i}{2}\rceil-1$ counts
the number of peaks appear before the $k$-th column. In particular, $n\in
I_{r}$ and from previous observation one has
$\lceil\frac{r}{2}\rceil=\operatorname{pk}w+1$. By (5.14) the total number of
bars and marked columns before the $n$th column is
$|f(w_{n})|-\operatorname{pk}w-1-\delta_{n}$. Together with the fact that the
$n$th column is marked if and only if $\delta_{n}+\gamma_{n}=1$ and $n\in{\cal
M}_{w}$, the total number of bars that should be added after the $n$th column
is

$m-\operatorname{pk}w-1-(|f(w_{n})|-\operatorname{pk}w-1)-\delta_{n}+(\delta_{n}+\gamma_{n})\cdot\delta(n\in{\cal
M}_{w})=m-|f(w_{n})|+\delta_{n}-(\delta_{n}+\gamma_{n})\cdot\delta(n\in{\cal
M}_{w}).$

Hence the nonnegativity condition becomes the following inequality,

$m-|f(w_{n})|+\delta_{n}-(\delta_{n}+\gamma_{n})\cdot\delta(n\in{\cal
M}_{w})\geq 0,$

or equivalently,

$m-|f(w_{n})|\geq(\delta_{n}+\gamma_{n})\cdot\delta(n\in{\cal
M}_{w})-\delta_{n}.$ (5.17)

By assumption $f$ has absolute value at most $m$, hence $m-|f(w_{n})|\geq 0$.
Therefore, one only needs to consider whether the inequality holds when
$(\delta_{n}+\gamma_{n})\cdot\delta(n\in{\cal M}_{w})-\delta_{n}=1$, in which
case it means $\delta_{n}+\gamma_{n}=1$, $\delta(n\in{\cal M}_{w})=1$ and
$\delta_{n}=0$, or equivalently, $n\in I_{i}$ where $i$ is odd and $n\in{\cal
M}_{w}$. This is a contradiction as $n+1$ must be in a run where the index is
even by definition, which implies that $n$ and $n+1$ cannot be in the same run
index set, hence $n\notin{\cal M}_{w}$. Now it is clear that
$(\delta_{n}+\gamma_{n})\cdot\delta(n\in{\cal M}_{w})-\delta_{n}\leq 0$, which
completes the proof of inequality (5.17). It therefore follows that the
marking we constructed is a $(w,m)$-marking.

Now we need to show that for a given $(w,m)$-marking, there are
$2^{2\operatorname{pk}w+1}$ different associated functions. From the above
construction we notice that for those $k\in{\cal M}_{w}$, $f(w_{k})$ is
uniquely determined. More precisely, whether column $k$ gets marked determines
$\delta_{k}$ and $\gamma_{k}$ as $k$ determines the parity of $i$, hence the
sign of $f(w_{k})$ by the definition of $\delta_{k}$ and $\gamma_{k}$. Suppose
$k\in I_{i}$. The number of marks and bars strictly before the $k$th column
determines the value $|f(w_{k})|-\lceil i/2\rceil-\delta_{k}$, therefore it
determines $f(w_{k})$ as well. The only ambiguity is about $f(w_{k})$ for
those $k\notin{\cal M}_{w}$. As the number of marks and bars strictly before
the $k$th column fixes the value $|f(w_{k})|-\lceil i/2\rceil-\delta_{k}$,
assuming to be $L$, there are two possible choices of the value $f(w_{k})$ for
each $k\notin{\cal M}_{w}$: if $i$ is even, then $f(w_{k})=-(L+\lceil
i/2\rceil+1)$ or $f(w_{k})=L+\lceil i/2\rceil$; if $i$ is odd, then
$f(w_{k})=L+\lceil i/2\rceil$ or $f(w_{k})=-(L+\lceil i/2\rceil)$. It follows
that there are $2^{2\operatorname{pk}w+1}$ different functions corresponding
to a given $(w,m)$-marking. ∎

######  Example 5.17.

Still consider the permutation $w=143256$, and an enriched $w$-partition $f$
defined as follows:

$f(1)=1,\;f(4)=-2,\;f(3)=-4,\;f(2)=-4,\;f(5)=-5,\;f(6)=5.$

The corresponding marking is the first one in the Example 5.15.

We now give a totally combinatorial derivation of the order polynomial
$\Omega^{\operatorname{cyc}}([w],m)$ by using the Proposition 5.16, which is
also proved in a combinatorial way.

######  Corollary 5.18.

For a given $[w]\in[\mathcal{S}_{n}]$, one has

$\displaystyle\Omega^{\operatorname{cyc}}([w],m)$
$\displaystyle=(n-2\operatorname{cpk}w)\cdot
2^{2\operatorname{cpk}w+1}\sum_{k=0}^{m-1-\operatorname{cpk}w}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{cpk}w-1}{m-1-\operatorname{cpk}w-k}$
$\displaystyle\qquad+\operatorname{cpk}w\cdot
2^{2\operatorname{cpk}w}\sum_{k=0}^{m-\operatorname{cpk}w}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{cpk}w+1}{m-\operatorname{cpk}w-k}.$

This right side of the equation does not depend on the choice of
representative $w$, as they all have the same cyclic peak number.

###### Proof.

Notice that any representative $w^{\prime}$ of $[w]$ satisfies
$\operatorname{pk}w^{\prime}=\operatorname{cpk}[w]-1$ if $w^{\prime}$ starts
or ends with a cyclic peak. Otherwise,
$\operatorname{pk}w^{\prime}=\operatorname{cpk}[w]$. Among the $n$
representatives of $[w]$, there are $2\operatorname{cpk}[w]$ of them with a
peak element at one end. Therefore, applying equation (5.4) we have

$\Omega^{\operatorname{cyc}}([w],m)=\Delta^{\operatorname{cyc}}_{[w]}(1^{m})=\sum_{v\in[w]}\Delta_{v}(1^{m})=\sum_{v\in[w]}\Omega(v,m),$

and by the previous observation and the Proposition 5.16, one has

$\displaystyle\sum_{v\in[w]}\Omega(v,m)$
$\displaystyle=\sum_{v\in[w]}2^{2\operatorname{pk}v+1}\sum_{k=0}^{m-1-\operatorname{pk}v}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{pk}v-1}{m-1-\operatorname{pk}v-k}$
$\displaystyle=(n-2\operatorname{cpk}[w])\cdot
2^{2\operatorname{cpk}[w]+1}\sum_{k=0}^{m-1-\operatorname{cpk}[w]}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{cpk}[w]-1}{m-1-\operatorname{cpk}[w]-k}$
$\displaystyle\quad+2\operatorname{cpk}[w]\cdot
2^{2(\operatorname{cpk}[w]-1)+1}\sum_{k=0}^{m-1-(\operatorname{cpk}[w]-1)}\left(\binom{n+1}{k}\right)\binom{n-2(\operatorname{cpk}[w]-1)-1}{m-1-(\operatorname{cpk}[w]-1)-k}$
$\displaystyle=(n-2\operatorname{cpk}w)\cdot
2^{2\operatorname{cpk}w+1}\sum_{k=0}^{m-1-\operatorname{cpk}w}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{cpk}w-1}{m-1-\operatorname{cpk}w-k}$
$\displaystyle\qquad+\operatorname{cpk}w\cdot
2^{2\operatorname{cpk}w}\sum_{k=0}^{m-\operatorname{cpk}w}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{cpk}w+1}{m-\operatorname{cpk}w-k}$

The conclusion follows immediately. ∎

We note that the generating function of order polynomials in Proposition 5.13
can also be deduced from the corollary above. More precisely,

$\displaystyle\sum_{m}\Omega^{\operatorname{cyc}}([w],m)t^{m}$
$\displaystyle=(n-2\operatorname{cpk}w)\cdot
2^{2\operatorname{cpk}w+1}\,\frac{(1+t)^{n-2\operatorname{cpk}w-1}}{(1-t)^{n+1}}\,t^{\operatorname{cpk}w+1}$
$\displaystyle\quad+\operatorname{cpk}w\cdot
2^{2\operatorname{cpk}w}\,\frac{(1+t)^{n-2\operatorname{cpk}w+1}}{(1-t)^{n+1}}\,t^{\operatorname{cpk}w}$
$\displaystyle=\left(\frac{4t}{(1+t)^{2}}\right)^{\operatorname{cpk}w}\left(\frac{1+t}{1-t}\right)^{n+1}\left((n-2\operatorname{cpk}w)\cdot\frac{2t}{(1+t)^{2}}+\operatorname{cpk}w\right)$
$\displaystyle=\left(\frac{4t}{(1+t)^{2}}\right)^{\operatorname{cpk}w}\left(\frac{1+t}{1-t}\right)^{n-1}\left(\operatorname{cpk}w+\frac{2nt}{(1-t)^{2}}\right).$

In [GZ18], Ira and Zhuang discussed shuffle-compatible permutation statistics
in terms of shuffle algebras. In the Theorem 4.8, they proved that peak number
$\operatorname{pk}$ is shuffle compatible and also characterized its shuffle
algebra ${\cal A}_{\operatorname{pk}}$. It turns out that the Proposition 5.16
gives another characterization of ${\cal A}_{\operatorname{pk}}$. Before we
state and prove the theorem, we review some definitions and results from
[GZ18].

Suppose $\pi$ and $\sigma$ are two disjoint permutations of length $m$ and $n$
respectively. Then the shuffle set of $\pi$ and $\sigma$ is

$\pi\shuffle\sigma=\\{\tau:\text{$|\tau|=m+n$, $\pi$ and $\sigma$ are
subsequences of $\tau$}\\}.$

A statistic $\operatorname{st}$ is (linear) shuffle compatible if for disjoint
permutations $\pi$ and $\sigma$, the multiset
$\\{\\{\,\operatorname{st}(\tau):\tau\in\pi\shuffle\sigma\,\\}\\}$ only
depends on $\operatorname{st}(\pi)$, $\operatorname{st}(\sigma)$, $|\pi|$ and
$|\sigma|$.

Every linear permutation statistic $\operatorname{st}$ induces an equivalence
relation on permutations. More precisely, two permutations $\pi$ and $\sigma$
are $\operatorname{st}$-equivalent if
$\operatorname{st}(\pi)=\operatorname{st}(\sigma)$ and $|\pi|=|\sigma|$, and
the $\operatorname{st}$-equivalence class of $\pi$ is denoted by
$[\pi]_{\operatorname{st}}$. Moreover, if $\operatorname{st}$ is shuffle
compatible, one can associate to $\operatorname{st}$ a $\mathbb{Q}$-algebra as
follows: first we associate to $\operatorname{st}$ a $\mathbb{Q}$-vector space
by taking the $\operatorname{st}$-equivalence classes as a basis, then define
multiplication by

$[\pi]_{\operatorname{st}}[\sigma]_{\operatorname{st}}=\sum_{\tau\in\pi\shuffle\sigma}[\tau]_{\operatorname{st}}.$

Here the shuffle-compatibility of $\operatorname{st}$ guarantees that the
above multiplication is well-defined. In this case, we call the resulted
algebra the shuffle algebra of $\operatorname{st}$ and denote it by ${\cal
A}_{\operatorname{st}}$.

We recall the characterization of peak shuffle algebra ${\cal
A}_{\operatorname{pk}}$ from [GZ18] is:

######  Theorem 5.19 ([GZ18], Theorem 4.8 (b)).

The linear map on ${\cal A}_{\operatorname{pk}}$ defined by

$[\pi]_{\operatorname{pk}}\mapsto\begin{cases}\displaystyle\frac{2^{2\operatorname{pk}\pi+1}t^{\operatorname{pk}\pi+1}(1+t)^{|\pi|-2\operatorname{pk}\pi-1}}{(1-t)^{|\pi|+1}}x^{|\pi|},&\text{if
$|\pi|\geq 1$};\\\\[6.0pt] \displaystyle\frac{1}{1-t},&\text{if
$|\pi|=0$,}\end{cases}$

is a $\mathbb{Q}$-algebra isomorphism from ${\cal A}_{\operatorname{pk}}$ to
the span of

$\left\\{\frac{1}{1-t}\right\\}\bigcup\left\\{\frac{2^{2j+1}t^{j+1}(1+t)^{n-2j-1}}{(1-t)^{n+1}}x^{n}\right\\}_{n\geq
1,\,0\leq j\leq\lfloor\frac{n-1}{2}\rfloor}$

a subalgebra of $\mathbb{Q}[[t\ast]][x]$.∎

Now we give another characterization of ${\cal A}_{\operatorname{pk}}$ from
our Proposition 5.16.

######  Theorem 5.20.

The linear map on ${\cal A}_{\operatorname{pk}}$ defined by

$[\pi]_{\operatorname{pk}}\mapsto
2^{2\operatorname{pk}\pi+1}\sum_{k=0}^{m-1-\operatorname{pk}\pi}\left(\binom{|\pi|+1}{k}\right)\binom{|\pi|-2\operatorname{pk}\pi-1}{m-1-\operatorname{pk}\pi-k}x^{|\pi|}$

is a $\mathbb{Q}$-algebra isomorphism from ${\cal A}_{\operatorname{pk}}$ to
the span of

$\left\\{1\right\\}\bigcup\left\\{2^{2j+1}\sum_{k=0}^{m-1-j}\left(\binom{n+1}{k}\right)\binom{n-2j-1}{m-1-j-k}x^{n}\right\\}_{n\geq
1,\,0\leq j\leq\lfloor\frac{n-1}{2}\rfloor}$

a subalgebra of $\mathbb{Q}[x]^{\mathbb{N}}$, the algebra of functions
$\mathbb{N}\to\mathbb{Q}[x]$ in the non-negative integer value $m$.

###### Proof.

Define a map $\kappa_{m}:\operatorname{QSym}\to\mathbb{Q}[x]$ first by

$\kappa_{m}(F_{L})=K_{\operatorname{Pk}(L)}(1^{m})x^{n},$

and then linearly extend to the whole $\operatorname{QSym}$. The following
equation, [Ste97] equation (3.1),

$K_{\operatorname{Pk}\pi}\cdot
K_{\operatorname{Pk}\sigma}=\sum_{\tau\in\pi\shuffle\sigma}K_{\operatorname{Pk}\tau},$

implies that $\kappa_{m}$ is a $\mathbb{Q}$-algebra homomorphism. So the map
that takes $F_{L}$ to the function $\theta_{L}:m\mapsto\kappa_{m}(F_{L})$ is a
homomorphism from $\operatorname{QSym}$ to $\mathbb{Q}[x]^{\mathbb{N}}$. It
follows from the Proposition 5.16 that

$\kappa_{m}(F_{L})=2^{2\operatorname{pk}(L)+1}\sum_{k=0}^{m-1-\operatorname{pk}(L)}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{pk}(L)-1}{m-1-\operatorname{pk}(L)-k}x^{n}.$

Moreover, from Theorem 5.12 one has

$\sum_{m=0}^{\infty}2^{2\operatorname{pk}(L)+1}\sum_{k=0}^{m-1-\operatorname{pk}(L)}\left(\binom{n+1}{k}\right)\binom{n-2\operatorname{pk}(L)-1}{m-1-\operatorname{pk}(L)-k}t^{m}=\frac{1}{2}\left(\frac{1+t}{1-t}\right)^{n+1}\left(\frac{4t}{(1+t)^{2}}\right)^{1+\operatorname{pk}(L)},$

which is the generating function of $\theta_{L}$ and only depends on $n$ and
$\operatorname{pk}L$. They are clearly linearly independent for $L$ with
distinct peak numbers. The result follows immediately from [GZ18] Theorem 4.3.
∎

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