\documentclass[finalversion]{FPSAC2021}
\articlenumber{49}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
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\title{Specializations of colored quasisymmetric functions}
\author{Vassilis-Dionyssis Moustakas\thanks{\href{mailto:vasmous@math.uoa.gr}{vasmous@math.uoa.gr}. This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme "Human Resources Development, Education and Lifelong Learning" in the context of the project "Strengthening Human Resources Research Potential via Doctorate Research; 2nd Cycle" (MIS-5000432), implemented by the State Scholarships Foundation (IKY).}}
\address{Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece}
\received{November 30, 2020}

%% leave this blank until submitting a revised version
\revised{April 18, 2021}

\abstract{We propose a unified approach to prove general formulas for the joint distribution of an Eulerian statistic, a Mahonian statistic and the color sum statistic over a set of colored permutations by specializing Poirier's colored quasisymmetric functions. We apply this method to derive formulas for joint Euler--Mahonian and color sum distributions on colored permutations, derangements and involutions. A number of known formulas are recovered as special cases of our results, including formulas of Biagioli--Zeng, Assaf, Haglund--Loehr--Remmel, Biagioli--Caselli, Faliharimalala--Zeng and D\'{e}sarm\'{e}nien--Foata. Several new results are also obtained.}


%% put your keywords here, or comment this out if you don't have them yet
\keywords{Quasisymmetric function, generating function, specialization, descent set, major index, Euler--Mahonian distribution, Eulerian polynomial}

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%% defining operators
\def\newop#1{\expandafter\def\csname #1\endcsname{\mathop{\rm
#1}\nolimits}}
\newop{Des}
\newop{des}
\newop{ldes}
\newop{fdes}
\newop{maj}
\newop{fmaj}
\newop{csum}
\newop{fix}
\newop{eul}
\newop{mah}
\newop{ps}

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\newcommand{\fS}{\mathfrak{S}}
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\newcommand{\bx}{{\mathbf x}}
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\begin{document}

\maketitle

%% Introduction and motivation
\section{Introduction}
\label{sec: intro}

For a positive integer $n$, let $\fS_n$ be the set of permutations of $[n] := \{1, 2, \dots, n\}$. For $w \in \fS_n$, an index $i \in [n-1]$ is called a \emph{descent} of $w$, if $w(i) > w(i+1)$. The set of all descents of $w$, written $\Des(w)$, is called the \emph{descent~set} of $w$. The cardinality and the sum of all elements of $\Des(w)$ are written as $\des(w)$ and $\maj(w)$, respectively, and called the \emph{descent~number} and \emph{major~index} of $w$. A statistic on $\fS_n$ which is equidistributed with $\des$ (resp. $\maj$) is called \emph{Eulerian} (resp. \emph{Mahonian}). Let
\[
A_n (x, q) \ := \ \sum_{w \in \fS_n} \, x^{\des(w)} q^{\maj(w)}
\]
be the generating polynomial for the joint distribution $(\des, \maj)$ on $\fS_n$. The polynomial $A_n (x) := A_n (x, 1)$ is called  the $n$-th \emph{Eulerian~polynomial} and constitutes one of the most important polynomials in combinatorics (see, for example, \cite{Ath18}). 
	
It follows from the work of MacMahon \cite{MacCA} that
%
\begin{align}
\sum_{m \ge 0} \, [m + 1]_q^n \, x^m \ &= \ \frac{A_n (x, q)}{(1-x)(1-xq)\cdots (1-xq^n)} \label{eq: Carlitz} \\
A_n (1, q) \ &= \ [1]_q[2]_q\cdots[n]_q, \label{eq: MacMahon}
\end{align}
%
where $[n]_{q} := 1 + q + \cdots + q^{n-1}$ is the $q$-analogue of $n$. These formulas serve as the basis for many generalizations to Coxeter groups and $r$-colored permutation groups, that is wreath products $\ZZ_r \wr \fS_n$. Identities such as \cref{eq: Carlitz}, involving Euler--Mahonian distributions will be called \emph{Euler--Mahonian~identities}. As mentioned in \cite[Section~1]{BB13}, a general approach to prove Euler--Mahonian identities, among others, is via the theory of symmetric/quasisymmetric functions. We illustrate this approach by proving \cref{eq: Carlitz,eq: MacMahon} in a unified way, by specializing Gessel's fundamental quasisymmetric functions \cite[Section~7.19]{StaEC2}. This proof serves as a prototype for the proofs of all applications presented here (see \cref{sec: app}).  

 Let $\bx = (x_1, x_2, \dots)$ be a sequence of commuting indeterminates. The  \emph{fundamental~quasisymmetric~function} associated to $S \subseteq [n-1]$  is defined by  
%
\begin{equation} \label{eq:defFS(x)}
  F_{n, S} (\bx) \ :=
  \sum_{\substack{ i_1 \ge i_2 \ge \cdots \ge i_n \ge 1  \\ j \in S \,\Rightarrow\, 
  i_j > i_{j+1}}}
  x_{i_1} x_{i_2} \cdots x_{i_n}.
\end{equation}
%
The original definition actually defines $F_{n,S}(\bx)$ as in \cref{eq:defFS(x)} with $1 \le i_1 \le i_2 \le \cdots \le i_n$ instead. Our choice of definition will become clear in \cref{sec: special}. The \emph{stable~principal~specialization}, written $\ps$, and  \emph{principal~specialization} of order $m$,  written $\ps_m$, of a formal power series in $\bx$ are defined \cite[Section~7.8]{StaEC2} by the substitutions $x_i = q^{i-1}$, for all $i \ge 1$ and 
\[
x_i \ = \ 
\begin{cases}
q^{i-1}, &\, \text{if $1 \le i \le m$} \\
0, &\, \text{if $i > m$},
\end{cases}
\]
respectively. 

The principal specialization of order $m$ and stable principal specialization of the quasisymmetric generating function of a subset $\aA \subseteq \fS_n$ are given by the following formulas \cite[Theorem~5.3]{GR93}
%
\begin{align}
\sum_{m \ge 1} \, \ps_m F (\aA; \bx)  \, x^{m-1} \ &= \
\frac{\sum_{w \in \aA} \, x^{\des(w)} \, q^{\maj(w)}}{(1-x)(1-xq)\cdots (1-xq^n)} \label{eq: psmF(A;x)} \\
\ps F (\aA; \bx)  \ &= \ \frac{\sum_{w \in \aA} \, q^{\maj(w)}}{(1-q)(1-q^2) \cdots (1 - q^n)}, \label{eq: psF(A;x)}
\end{align}
%
where $F(\aA; \bx) := \sum_{w \in \aA} F_{n, \Des(w)}(\bx)$. The quasisymmetric function $F(\fS_n; \bx)$ is known to have \cite[Corollary~7.12.5]{StaEC2} the following nice form
%
\begin{equation}
\label{eq: qsym generating function of Sn}
F(\fS_n; \bx) = (x_1 + x_2 + \cdots)^n.
\end{equation}
%
The proof of \cref{eq: Carlitz,eq: MacMahon} follows by taking the principal specialization of order $m$  and the stable principal specialization of \cref{eq: qsym generating function of Sn} and substituting  in \cref{eq: psmF(A;x),eq: psF(A;x)} for $\aA = \fS_n$, respectively.

 In our recent paper \cite{Mou20}, we provide a unified approach to prove Euler--Mahonian identities on sets of colored permutations by specializing Poirier's colored analogue of quasisymmetric functions \cite{PoiThesis}. A first instance of this technique appears in the work of Athanasiadis \cite[Equation~(45)]{Ath18}. Our choice of colored quasisymmetric functions, and much of the motivation behind this paper comes from the fact that Poirier's signed analogue of the fundamental quasisymmetric functions was recently employed by Adin et al. \cite{AAER17} in order to define and study a signed analogue of the concept of fine sets and fine characters of Adin and Roichman (see, for example, \cite[Section~1]{AAER17}).
	
 This extended abstract is a summary of our results in \cite{Mou20} with applications involving color sum Euler--Mahonian identities. In \cref{sec: prelim}, we review colored permutation statistics and recall the definition of fundamental colored quasisymmetric functions. In \cref{sec: special}, we present the main results on specializations of colored quasisymmetric functions and in \cref{sec: app}, we apply these results to prove formulas for the joint Euler--Mahonian and color sum distributions on colored permutations, derangements and absolute involutions. 
 
%A number of known formulas are recovered as special cases of our results, including formulas of Biagioli--Zeng, Assaf, Haglund--Loehr--Remmel, Biagioli--Caselli, Faliharimalala--Zeng and D\'{e}sarm\'{e}nien--Foata.

%% Preliminaries
\section{Preliminaries}
\label{sec: prelim}

This section provides key definitions regarding colored permutation statistics and colored quasisymmetric functions. For a survey on Euler--Mahonian identities for the colored permutation groups, we refer the reader to \cite[Section~2.1]{Mou20}. We assume familiarity with basic concepts in the theory of symmetric functions as presented in \cite[Section~7]{StaEC2}. For nonnegative integers $0 \le k \le n$, define 
%
\begin{align*} 
(x; q)_n \ &:= \ \begin{cases} 1, & \text{if $n = 0$} \\
(1-x)(1-xq)\cdots(1-xq^{n-1}), & \text{if $n \ge 1$} 
\end{cases} \\
\binom{n}{k}_q \ &:= \ \begin{cases} 1, & \text{if $n = 0$} \\
\dfrac{[n]_q!}{[k]_q![n-k]_q!}, & \text{if $n \ge 1$} 
\end{cases} 
\end{align*}
%
and set $(q)_n := (q; q)_n$. 

\subsection{Colored permutation statistics}
\label{subsec: color}

Fix a positive integer $r$ and let 
\[
\Om_{n, r} := \{1^0, 2^0, \dots, n^0, 1^1, 2^1, \dots, n^1, \dots, 1^{r-1}, 2^{r-1}, \dots, n^{r-1}\}
\]
be the set of \emph{$r$-colored integers}. We will often identify colored integers $i^0$ with $i$. The \emph{$r$-colored~permutation~group}, denoted by $\fS_{n, r}$, consists of all permutations $w$ of $\Om_{n, r}$ such that $w (a^0)=b^j \toto w (a^i)=b^{i +j}$, where $i + j$ is computed modulo $r$ and the product of $\fS_{n, r}$ is composition of permutations.  The $r$-colored permutation group can be realized as the wreath product group $\ZZ_r \wr \fS_n$, where the elements of $\ZZ_r$ are thought of as colors. The elements of $\fS_{n, r}$ are represented  in window notation as $w = w(1)^{c_1}w(2)^{c_2}\cdots w(n)^{c_n}$, where $w(1)w(2)\cdots w(n) \in \fS_n$ is the \emph{underlying~permutation} and $(c_1, c_2, \dots, c_n)$ is the \emph{color~vector} of $w$. We will represent both the colored permutation and the underlying permutation by the same letter.

Consider the following total order
\[
1^{r-1} <_c \cdots <_c n^{r-1} <_c \cdots <_c 1^1 <_c \cdots <_c n^1 <_c  1^0 <_c \cdots <_c n^0
\]
on $\Om_{n, r}$, sometimes called the \emph{color order}. For $w \in \fS_{n, r}$, define $\Des(w)$ to be the set of all indices $i \in [n-1]$ such that  $w(i) >_c w(i+1)$ together with 0, whenever $w(1)$ has nonzero color and let $\des(w)$ be its cardinality. Write $\Des^*(w) := \Des(w) \sm \{0\}$ and $\des^*(w)$ for its cardinality. Also, define
%
\begin{align*}
\maj(w) \ &:= \ \sum_{i \in \Des^*(w)} i \\
\csum(w) \ &:= \ c_1 + c_2 + \cdots + c_n \\
\fmaj(w) \ &:= \ r\maj(w) + \csum(w) \\
\fdes(w) \ &:= \ r\des^*(w) + c_1,
\end{align*}
%
the \emph{major index, color sum, flag major index} and \emph{flag descent number} of $w$, respectively. For a complete account of these statistics we refer to \cite[Section~2.1]{Mou20}. 

For a pair of statistics $(\eul, \mah)$ on $\fS_{n, r}$ and $\aA \subseteq \fS_{n, r}$, we define the following generating polynomials
%
\begin{align*}
\aA_{n, r}^{(\eul, \mah, \csum)}(x, q, p) \ &:=
\  \sum_{w \in \aA} \, x^{\eul(w)} q^{\mah(w)} p^{\csum(w)} \\
\aA_{n, r}^{(\mah, \csum)}(q, p) \ &:= \ \aA_{n, r}^{\eul, \mah, \csum}(1, q, p) \\
\aA_{n, r}^{(\eul, \csum)}(x, p) \ &:= \ \aA_{n, r}^{\eul, \mah, \csum}(x, 1, p) \\
\aA_{n, r}^{(\eul, \mah)}(x, q) \ &:= \ \aA_{n, r}^{\eul, \mah, \csum}(x, q, 1).
\end{align*}
%

\subsection{Colored quasisymmetric functions}
\label{subsec: colored qsym}

Let $\bx^{(j)} = (x_1^{(j)}, x_2^{(j)}, \dots)$ be sequences of commuting indeterminates, for every $0 \le j \le r-1$. We consider formal power series in $\bx^{(0)}, \bx^{(1)}, \dots, \bx^{(r-1)}$ with complex coefficients. The \emph{fundamental colored quasisymmetric function} associated to $w \in \fS_{n, r}$ with color vector $(c_1, c_2,  \dots, c_n)$ is defined by
%
\begin{equation}
\label{eq: fundamental colored quasi}
F_w (\bx^{(0)}, \bx^{(1)}, \dots, \bx^{(r-1)}) \ := \ 
\sum_{\substack{i_1 \ge i_2 \ge \cdots \ge i_n \ge 1 \\ j \in \Des^*(w) \ \toto \  i_j > i_{j+1}}} \, x_{i_1}^{(c_1)} x_{i_2}^{(c_2)} \cdots x_{i_n}^{(c_n)}.
\end{equation}
%

This colored analogue of Gessel's fundamental quasisymmetric functions was introduced by Poirier \cite{PoiThesis} and further studied in \cite{HP10}  and more recently, for $r=2$, in \cite{AAER17}. One can define (colored) quasisymmetric functions indexed by \emph{colored compositions}, that is compositions of a positive integer whose parts are colored. This is the approach of \cite[Section~3]{HP10}. Our definition is slightly different, but equivalent to, the one given in \cite{HP10} with the inequalities under the summation on the right-hand side of \cref{eq: fundamental colored quasi} being reversed.

Lastly, for a subset $\aA \subseteq \fS_{n, r}$,  let 
\[
F (\aA; \bx^{(0)}, \bx^{(1)}, \dots, \bx^{(r-1)}) \ := \ \sum_{w \in \aA} \, F_w (\bx^{(0)}, \bx^{(1)}, \dots, \bx^{(r-1)})
\]
be the colored quasisymmetric generating function associated to $\aA$. 

%% Main results
\section{Specializations}
\label{sec: special}

This section provides general formulas for the generating polynomials of joint Mahonian and color sum statistics and joint Euler--Mahonian and color sum statistics on sets of colored permutations by specializing fundamental colored quasisymmetric functions. We first state the results and at the end of the section comment on their proofs, the complete versions of which can be found at \cite[Section~3]{Mou20}.

We begin by considering the specialization $\ps_{q, p}^{(r)}$ defined by the substitutions $
x_i^{(j)} = q^{i-1} p^j$ for every $i \ge 1$ and $0 \le j \le r-1$, the specialization $\ps_{q, p, m}^{(r)}$ defined by the substitutions
\[
\begin{cases}
x_i^{(0)} = q^{i-1}, \ &\quad 1 \le i \le m   \\
x_i^{(j)} = q^{i-1} p^j, \ &\quad \text{$1 \le j \le r-1$ and $1 \le i \le m-1$} \\
x_i^{(j)} = 0, \ &\quad \text{otherwise}
\end{cases}
\]
and the specialization $\wtl{\ps}_{q, p, m}^{(r)}$, defined by the substitutions $x_i^{(j)} = q^{i-1} p^j$ for every $1 \le i \le m$ and $0 \le j \le r-1$ and $x_i^{(j)} = 0$ otherwise. 
%
\begin{theorem}
\label{thm: I}
For a positive integer $n$ and every $\aA \subseteq \fS_{n, r}$, we have 
%
\begin{align}
\ps_{q, p}^{(r)} F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \ &= \ \frac{\aA_{n, r}^{(\maj, \csum)}(q, p)}{(q)_n} \label{eq: Imaj}\\ 
\sum_{m \ge 1} \, \ps_{q, p, m}^{(r)}F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \, x^{m-1} \ &= \ \frac{\aA_{n, r}^{(\des, \maj, \csum)}(x, q, p)}{(x; q)_{n +1}} \label{eq: Idesmaj}\\
\sum_{m \ge 1} \, \wtl{\ps}_{q, p, m}^{(r)} F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \, x^{m-1} \ &= \ \frac{\aA_{n, r}^{(\des^*, \maj, \csum)}(x, q, p)}{(x; q)_{n +1}}. \label{eq: Ides*maj}
\end{align}
%
\end{theorem} 
%

Next, we consider the specialization $\psi_{q, p}^{(r)}$ defined by the substitutions $x_i^{(j)} = q^{r(i-1) + j} p^j$ for every $i \ge 1$ and $0 \le j \le r-1$,  the specialization $\psi_{q, p, m}^{(r)}$ defined by the substitutions 
\[
\begin{cases}
x_i^{(0)} = q^{r(i-1)}, \ &\quad 1 \le i \le m   \\
x_i^{(j)} = q^{r(i-1) + j} p^j, \ &\quad \text{$1 \le j \le r-1$ and $1 \le i \le m-1$} \\
 x_i^{(j)} = 0, \ &\quad \text{otherwise}
\end{cases}
\]
and the specialization $\wtl{\psi}_{q, p, m}^{(r)}$ defined by the substitutions $x_i^{(j)} = q^{r(i-1) + j} p^j$ for every $1 \le i \le m$ and $0 \le j \le r-1$ and $x_i^{(j)} = 0$ otherwise.
%
\begin{theorem}
\label{thm: II}
For a positive integer $n$ and every $\aA \subseteq \fS_{n, r}$, we have 
%
\begin{align}
\psi_{q, p}^{(r)} F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \ &= \ \frac{\aA^{(\fmaj, \csum)}(q, p)}{(q^r)_n} \label{eq: IIfmaj}\\ 
\sum_{m \ge 1} \, \psi_{q, p, m}^{(r)} F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \, x^{m-1}  \ &= \ \frac{\aA^{(\des, \fmaj, \csum)}(x, q, p)}{(x; q^r)_{n +1}} \label{eq: IIdesfmaj}\\
\sum_{m \ge 1} \, \wtl{\psi}_{q, p, m}^{(r)} F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \, x^{m-1} \ &= \ \frac{\aA^{(\des^*, \fmaj, \csum)}(x, q, p)}{(x; q^r)_{n +1}}. \label{eq: IIdes*fmaj}
\end{align}
%
\end{theorem}
%

We remark that \cref{thm: II} follows from \cref{thm: I} by setting $q \to q^r$ and $p \to qp$. Lastly, we consider a more complicated specialization $\phi_{q, p, m}^{(r)}$ defined as follows: If $m = rs + t$, for some $1 \le t \le r$ and $s \ge 0$, then let $x_i^{(j)} = 0$ if the pair $(i, j)$ is lexicographically greater than the pair $(m - t + 1, t -1)$ and otherwise
\[
x_i^{(j)} \ = \ 
\begin{cases}
q^{i + j - 1} p^j, &\, \text{if $i \equiv 1 \pmod{r}$} \\
0, &\, \text{if $i \not\equiv 1 \pmod{r}$}.
\end{cases}
\] 

We illustrate the definition of $\phi_{q, p, m}^{(r)}$ by considering a specific example for $r = 3$ and $m \in \{7, 8, 9\}$. The substitutions become
\[
(x_i^{(j)} )_{\substack{0 \le j \le 2 \\ i \ge 1}} \ = \
\begin{cases}
\begin{pmatrix}
1 & 0 & 0 & q^3 & 0 & 0 & q^6 & 0 & \cdots\\
q p & 0 & 0 & q^4 p & 0 & 0 & 0 & 0 & \cdots  \\
q^2 p^2 & 0 & 0 & q^5 p^2  & 0 & 0 & 0 & 0 & \cdots \\
\end{pmatrix}, 
&\, \text{if $m = 7$} \\
\begin{pmatrix}
1 & 0 & 0 & q^3 & 0 & 0 & q^6 & 0 & \cdots \\
q p & 0 & 0 & q^4 p & 0 & 0& q^7p & 0 & \cdots \\
q^2 p^2 & 0 & 0 & q^5 p^2  & 0 & 0 & 0 & 0 & \cdots \\
\end{pmatrix},
&\, \text{if $m = 8$} \\
\begin{pmatrix}
1 & 0 & 0 & q^3 & 0 & 0 & q^6 & 0 & \cdots \\
q p & 0 & 0 & q^4 p & 0 & 0& q^7p & 0 & \cdots \\
q^2 p^2 & 0 & 0 & q^5 p^2  & 0 & 0 &  q^8 p^2 & 0 & \cdots \\
\end{pmatrix},
&\, \text{if $m = 9$}. 
\end{cases}
\]
%
\begin{theorem}
\label{thm: III}
For a positive integer $n$ and every $\aA \subseteq \fS_{n, r}$, we have 
%
\begin{equation}
\label{eq: IIIfdesfmaj}
\sum_{m \ge 1} \, \phi_{q, p, m}^{(r)} F (\aA; \bx^{(0)}, \dots, \bx^{(r-1)}) \, x^{m-1} \ = \ \frac{\aA^{(\fdes, \fmaj, \csum)}(x, q, p)}{(1-x)(1 - x^r q^r)(1 - x^r q^{2r}) \cdots (1 - x^r q^{nr})}.
\end{equation}
%
\end{theorem}
%

	The proofs of \Cref{thm: I,thm: III} are similar to each other and similar to those of \cref{eq: psF(A;x),eq: psmF(A;x)} and can be found in \cite[Section~3]{Mou20}. The key observation in the proof of \cref{eq: Idesmaj}, for example, is the following identity
%
\begin{equation}
\label{eq: key}
 \ps_{q, p, m}^{(r)} F_w (\bx^{(0)}, \dots, \bx^{(r-1)}) \ = \ \sum_{\substack{m := i_0 \ge i_1 \ge \cdots \ge i_n \ge 1 \\ j \in \Des(w) \ \toto \ i_j > i_{j+1}}} q^{i_1 + \cdots + i_n - n + \csum(w)} p^{\csum(w)}.
\end{equation}
%
This is the case, because under the specialization $\ps_{q, p, m}^{(r)}$ substitutions $x_m^{(1)}, x_m^{(2)}, \dots, x_m^{(r-1)}$ occur only if the color of the first entry of $w$ is nonzero, which is exactly when $0$ is considered a descent of $w$, explaining the first inequality under the sum on the right-hand side of \cref{eq: key}. Lastly, notice that our choice of the direction of inequalities in the definition of $F_w$ (see \cref{eq: fundamental colored quasi}) allows us to deal with the major index instead of the comajor index, as done in \cite[Lemma~5.2]{GR93} for example. This explains the motivation behind our choice.

%% Applications
\section{Applications; Color sum Euler--Mahonian identities on colored permutations, derangements and absolute involutions}
\label{sec: app}

This section applies the theorems of \cref{sec: special} to prove color sum Euler--Mahonian identities on colored permutations, derangements and absolute involutions
. 
%The strategy is that if the quasisymmetric generating function of a set of colored permutations $\aA$ can be specialized in a nice way, then a color sum Euler--Mahonian (and Mahonian) identity on $\aA$ can automatically be obtained.

For a positive integer $n$, we have
%
\begin{equation}
\label{eq: qsym generating function of Snr}
F (\fS_{n, r}; \bx^{(0)}, \dots, \bx^{(r-1)}) \ = \ ( e_1(\bx^{(0)}) + \cdots + e_1(\bx^{(r - 1)}))^n,
\end{equation}
%
where $e_1(\bx^{(j)}) := \sum_{i \ge 1} x_i^{(j)}$, for every $0 \le j \le r-1$. 
This formula reduces to \cref{eq: qsym generating function of Sn} for $r=1$. It appears in \cite[Proposition~1.13]{PoiThesis} in a more general setting, but can also be proved using a colored version of the Robinson--Schensted correspondence and the Frobenius formula for $\fS_{n, r}$ (see \cite[Lemma~2.1]{Mou20}). The following corollary follows from \cref{thm: I} for $\aA = \fS_{n, r}$ by specializing \cref{eq: qsym generating function of Snr}.
%
\begin{corollary}
\label{cor: ISnr}
For a positive integer $n$, we have
%
\begin{equation}
\label{eq: Snr maj}
\fS_{n, r}^{(\maj, \csum)}(q, p) \ = \ [r]_{p}^n [n]_{q}!
\end{equation}
and
%
\begin{align}
\sum_{m \ge 0} \, ([m+1]_q + p[r-1]_p[m]_q)^n \, x^m \ &= \ \frac{\fS_{n, r}^{(\des, \maj, \csum)}(x, q, p)}{(x; q)_{n+1}} \label{eq: Snr desmaj}
\\ 
\sum_{m \ge 0} \, ([r]_p[m+1]_q)^n \, x^m \ &= \ \frac{\fS_{n, r}^{(\des^*, \maj, \csum)}(x, q, p)}{(x; q)_{n+1}}. \label{eq: Snr des*maj}
\end{align}
%
\end{corollary}
%

\Cref{eq: Snr maj,eq: Snr desmaj} are due to Assaf \cite[Equation~(13)]{Ass10} and Biagioli and Zeng \cite[Equation~(8.1)]{BZ11}, respectively. The following corollary follows from \cref{thm: II} for $\aA = \fS_{n, r}$ by specializing \cref{eq: qsym generating function of Snr}.
%
\begin{corollary}
\label{cor: IISnr}
For a positive integer $n$, we have 
%
\begin{equation}
\label{eq: Snr fmaj}
\fS_{n, r}^{(\fmaj, \csum)}(q, p) \ = \ [r]_{pq}^n [n]_{q^r}!
\end{equation}
%
and
%
\begin{align}
\sum_{m \ge 0} \, ([m+1]_{q^r} + pq[m]_{q^r}[r-1]_{pq})^n \, x^m \ &= \ \frac{\fS_{n, r}^{(\des, \fmaj, \csum)}(x, q, p)}{(x; q^r)_{n+1}} \label{eq: Snr desfmaj} \\ 
\sum_{m \ge 0} \, ([r]_{pq}[m+1]_{q^r})^n \, x^m \ &= \ \frac{\fS_{n, r}^{(\des^*, \fmaj, \csum)}(x, q, p)}{(x; q^r)_{n+1}}. \label{eq: Snr des*fmaj}
\end{align}
%
\end{corollary}

\Cref{eq: Snr fmaj} refines Haglund, Loehr and Remmel's formula \cite[Equation~(34)]{HLR05} for the distribution of the flag major index over colored permutations and  \cref{eq: Snr desfmaj} appears in the work of Biagioli and Caselli \cite[Theorem~5.2]{BC12}. 
%
\begin{remark}
\begin{enumerate}
\item[(a)] \Cref{eq: Snr desmaj,eq: Snr desfmaj} for $q=1$ become
%
\begin{equation}
\label{eq: Snr des}
\sum_{m \ge 0} (m[r]_p + 1)^{n} x^m \ = \ \frac{\fS_{n, r}^{(\des, \csum)}(x, p)}{(1-x)^{n+1}},
\end{equation}
%
which reduces to an identity of Brenti \cite[Equation~(12)]{Bre94} for $r=2$. In particular, the polynomial $\fS_{n, r}^{(\des, \csum)}(x, p) \ := \ \sum_{i = 0}^n \ a_{n, r, i}(p) x^i$ satisfies the formula
\[
(m[r]_p + 1)^n \ = \ \sum_{i =0}^n \ a_{n, r, i}(p) \binom{m + n - i}{n}
\]
and therefore has only real roots for every positive integer $n$ and every $p \ge 1$ (cf.\ \cite[Corollary~3.7]{Bre94}). Although this result may not be new, it served as a motivation to introduce the parameter $p$ which keeps track of the color sum. 
\item[(b)] Setting $p = x$ in \cref{eq: Snr des*maj,eq: Snr des*fmaj} yields  Euler--Mahonian idendities for the pairs $(\ldes, \maj)$ and $(\ldes, \fmaj)$
%
\begin{align*}
\sum_{m \ge 0} \, [m+1]_q^n \, x^m \ &= \ \frac{\fS_{n, r}^{(\ldes, \maj)}(x, q)}{(x; q)_{n+1}[r]_x^n} \\
\sum_{m \ge 0} \, [m+1]_{q^r}^n \, x^m \ &= \ \frac{\fS_{n, r}^{(\ldes, \fmaj)}(x, q)}{(x; q^r)_{n+1}[r]_{qx}^n},
\end{align*}
%
where the \emph{length descent number} of a colored permutation $w \in \fS_{n, r}$ is defined by \cite[Definition~5.1]{Bag04} 
\[
\ldes(w) := \des^*(w) + \csum(w).
\]
\end{enumerate}
\end{remark}
%

The following corollary appears in the work of Biagioli and Caselli \cite[Theorem~5.4]{BC12} and can be proved by taking the $\phi_{q, p, m}^{(r)}$ specialization of \cref{eq: qsym generating function of Snr} and substituting in \cref{thm: III} for $\aA = \fS_{n, r}$.
%
\begin{corollary}
\label{cor: IIISnr}
For a nonnegative integer $m$, we write $m = r {\qq}(m) + {\rr}(m)$ for some nonnegative integer ${\qq}(m)$ and $0 \le {\rr}(m) < r$. Then, we have
%
\begin{equation}
\label{eq: Snr fdesfmaj}
\begin{split}
\sum_{m \ge 0} \, ([{\qq}(m) +1]_{q^r} + pq[r-1]_{pq}&[{\qq}(m)]_{q^r} + pq^{r{\qq}(m) + 1}[{\rr}(m)]_{pq})^n \, x^m   \\ 
&= \ \frac{\fS_{n, r}^{(\fdes, \fmaj, \csum)}(x, q, p)}{(1-x)(1 - x^r q^r)(1 - x^r q^{2r}) \cdots (1 - x^r q^{nr})}.
\end{split}
\end{equation}
%
\end{corollary}
%

An element of $\fS_{n, r}$ without fixed points of zero color is called a \emph{colored~derangement}. Let $\dD_{n, r}$ be the set of all colored derangements in $\fS_{n, r}$. For a positive integer $n$, we have
%
\begin{equation}
\label{eq: qsym generating function of Dnr}
F(\dD_{n, r}; \bx^{(0)}, \dots, \bx^{(r-1)}) \ = \ 
\sum_{k = 0}^n \, (-1)^k e_k(\bx^{(0)}) (e_1(\bx^{(0)}) + \cdots + e_1(\bx^{(r-1)}))^{n -k},
\end{equation}
%
where $e_k(\bx^{(j)})$ is the $k$-th elementary symmetric function in $\bx^{(j)}$. This formula reduces to \cite[Theorem~8.1]{GR93} and \cite[Equation~(7.8)]{AAER17} for $r=1$ and $r=2$, respectively and can be proved by trivially generalizing Adin et al.'s argument in the proof of \cite[Theorem~7.3]{AAER17} for general $r$. The following corollary is due to Assaf  \cite[Theorem~3.2]{Ass10} and can be proved by taking the $\ps_{q, p}^{(r)}$ specialization of \cref{eq: qsym generating function of Snr} and substituting in \cref{eq: Imaj} for $\aA = \fS_{n, r}$.
%
\begin{corollary}
\label{cor: IDnr}
For a positive integer $n$, we have
%
\begin{equation}
\label{eq: Dnr maj}
\dD_{n, r}^{(\maj, \csum)}(q, p) \ = \ 
[r]_p^n [n]_q! \, \sum_{k = 0}^n \, (-1)^k \, \frac{q^{\binom{k}{2}}}{[r]_p^k [k]_q!}.
\end{equation}
%
\end{corollary}
%

The following corollary follows from \cref{thm: II} for $\aA = \dD_{n, r}$ by specializing \cref{eq: qsym generating function of Dnr}.

\begin{corollary}
For a positive integer $n$, we have 
%
\begin{equation}
\label{eq: Dnr fmaj}
\dD_{n, r}^{(\fmaj, \csum)}(q, p) \ = \ 
[r]_{pq}^n [n]_{q^r}! \, \sum_{k = 0}^n \, (-1)^k \, \frac{q^{r\binom{k}{2}}}{[r]_{pq}^k [k]_{q^r}!}
\end{equation}
%
and
%
\begin{equation}
\label{eq: Dnr desfmaj}
%
\begin{split}
\sum_{m \ge 0} \, \sum_{k = 0}^n \, (-1)^k q^{r\binom{k}{2}} \binom{m+1}{k}_{q^r} \, ([m+1]_{q^r} &+ pq[m]_{q^r}[r-1]_{pq})^{n - k} \, x^m  \\ 
&= \ \frac{\dD_{n, r}^{(\des, \fmaj, \csum)}(x, q, p)}{(x; q^r)_{n+1}}.
\end{split}
%
\end{equation}
%
\end{corollary}

\Cref{eq: Dnr fmaj} refines Faliharimalala and Zeng's formula \cite[Equation~(2.5)]{FZ11} for the distribution of the flag major index over colored derangements (see also \cite[Equation~(6.8)]{FH08}). \Cref{eq: Dnr desfmaj} appears to be new even in the case $r=1$, where in this case refines a celebrated result of Wachs \cite[Theorem~4]{Wa89}.  

For $w \in \fS_{n, r}$ with color vector $(c_1, \dots, c_n)$, let $\ol{w}$ be the $r$-colored permutation with underlying permutation $w$ and color vector $(-c_1, \cdots, -c_n)$. A colored permutation $w \in \fS_{n, r}$ such that $\ol{w}^{-1} = w$ is called an \emph{absolute involution}. Let $\iI_{n, r}$ be the set of all absolute involutions in $\fS_{n, r}$. For a discussion on colored involutions (elements $w \in \fS_{n, r}$ such that $w^{-1} = w$) and absolute involutions we refer to \cite[Section~4.3]{Mou20}. 
We have
%
\begin{equation}
\label{eq: qsym generating function if Inr}
\sum_{n \ge 0} \, F (\iI_{n, r}; \bx^{(0)}, \dots, \bx^{(r-1)}) \, z^n \ = \
\prod_{c = 0}^{r-1} \, \prod_{i \ge 1} \, (1 - z x_i^{(c)})^{-1} \, \prod_{1 \le i < j} \, (1 - z^2 x_i^{(c)} x_j^{(c)})^{-1}, 
\end{equation}
%
where $F(\iI_{0, r}; \bx^{(0)}, \dots, \bx^{(r-1)}) := 1$. This formula specializes to \cite[Equation~(7.1)]{GR93} and can be proved using a colored version of the Robinson--Schensted correspondence and \cite[Corollary~7.13.8]{StaEC2} (see \cite[Theorem~4.8]{Mou20}). 

\Cref{cor: Inr} follows from either \cref{eq: Idesmaj} or \cref{eq: IIdesfmaj} for $q=1$ and $\aA = \iI_{n, r}$ by specializing \cref{eq: qsym generating function if Inr}.

%
\begin{corollary}
\label{cor: Inr}
If
\[
\iI_{n, r}^{(\des, \fix, \csum)}(x, y, p) \ := \
\begin{cases}
\displaystyle\sum_{w \in \iI_{n, r}} \, x^{\des(w)} y^{\fix(w)} p^{\csum(w)}, &\, \text{if $n \ge 1$} \\
1, &\, \text{if $n = 0$},
\end{cases}
\]
where $\fix(w)$ is the number of fixed points of zero color of $w$, then we have
%
\begin{equation}
\label{eq: Inr desfix}
\sum_{n \ge 0} \, \frac{\iI_{n, r}^{(\des, \fix, \csum)}(x, y, p)}{(1-x)^{n+1}} \, z^n \ = \ 
\sum_{m \ge 0} \frac{x^m}{(1+z)^m(1-zy)^{m+1}(z; p)_r^m (z^2; p^2)_r^{\binom{m}{2}}}.
\end{equation}
%
\end{corollary}

 \Cref{eq: Inr desfix} reduces to an identity of D{\'e}sarm{\'e}nien and Foata \cite[Equation~(6.2)]{DF85} for $r=1$ and to \cite[Equation~(2)]{Mou19} for $r=2$ and $y=p=1$. We remark that one can prove similar formulas for the joint Euler--Mahonian and color sum distributions on absolute involutions.
 
 We close by mentioning that formulas for color sum bimahonian and bieulerian--bimahonian distributions can be proved using the method of this paper. We refer to \cite[Section~4.3]{Mou20} for more details.

\acknowledgements{The author would like to thank Christos Athanasiadis for sharing his ideas on specializations of quasisymmetric functions.}

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