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\newcommand\hmmax{0}
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\usepackage{mathabx,bm,mathtools,shuffle,stmaryrd}

\newcounter{cms}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{pro}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{con}[thm]{Conjecture}
\newtheorem{cor}[thm]{Corollary}
\newtheorem*{conn}{Conjecture}
\theoremstyle{definition}
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\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\GG}{\mathbb{G}}
\newcommand{\UU}{\mathbb{U}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\aA}{\mathcal{A}}
\newcommand{\bB}{\mathcal{B}}
\newcommand{\fS}{\mathfrak{S}}
\newcommand{\fB}{\mathfrak{B}}

\newcommand{\PP}{\mathfrak{P}}

\newcommand{\cA}{\mathcal{A}}

\newcommand{\cc}{\operatorname{cc}}
\newcommand{\bp}{{\bm p}}
\newcommand{\A}{{\mathcal A}}
\newcommand{\ve}{{\varepsilon}}
\newcommand{\ee}{{\varepsilon}}

\newcommand{\dup}{\%}

\newcommand{\Iverson}[1]{\llparenthesis{#1}\rrparenthesis}

\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}

\DeclareMathOperator{\stat}{stat}
\DeclareMathOperator{\st}{st}

\DeclareMathOperator{\GL}{GL}

\DeclareMathOperator{\Des}{Des}
\DeclareMathOperator{\sDes}{sDes}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\QSym}{QSym}
\DeclareMathOperator{\rmsum}{sum}

\DeclareMathOperator{\comaj}{comaj}
\DeclareMathOperator{\des}{des}
\DeclareMathOperator{\maj}{maj}


\DeclareMathOperator{\Mat}{M}

\DeclareMathOperator{\ask}{ask}
\DeclareMathOperator{\concnt}{cc}
\DeclareMathOperator{\oc}{oc}

\DeclareMathOperator{\palette}{{pal}}
\DeclareMathOperator{\symbols}{{sym}}
\DeclareMathOperator{\std}{{std}}
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%% define your title in the usual way
\title{From coloured permutations to Hadamard products and zeta functions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

\author[A.~Carnevale, V.~D.~Moustakas, T.~Rossmann]{
 Angela Carnevale\thanks{\href{mailto:angela.carnevale@universityofgalway.ie}{angela.carnevale@universityofgalway.ie}. AC was partially supported by the College of Science and Engineering at the University of Galway through a Strategic Research Fund.}\addressmark{1}, 
Vassilis Dionyssis Moustakas\thanks{\href{mailto:vasileios.moustakas@dm.unipi.it}{vasileios.moustakas@dm.unipi.it}}\addressmark{2},
\and 
Tobias~Rossmann\thanks{\href{mailto:tobias.rossmann@nuigalway.ie}{tobias.rossmann@universityofgalway.ie}}\addressmark{1}
}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1} School of Mathematical and  Statistical Sciences, University of Galway, Ireland \\
  \addressmark{2} Dipartimento di Matematica, Universit\`a di Pisa, Italy}

%% put the date of submission here
\received{15 November 2023}

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\revised{27 March 2024}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We devise a constructive method for computing 
  explicit combinatorial formulae for Hadamard products of
  certain rational generating functions.
  The latter arise naturally when studying so-called ask zeta functions of direct sums of
   modules of matrices or class- and orbit-counting zeta functions of
  direct products of groups.
  Our method relies on shuffle compatibility of coloured permutation
  statistics and coloured quasisymmetric functions, extending recent work of
  Gessel and Zhuang.
}

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%% put your keywords here, or comment this out if you don't have them yet
\keywords{Coloured permutations, permutation statistics, Hadamard products,
  shuffle compatibility, average sizes of kernels, zeta functions}

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\usepackage[backend=bibtex]{biblatex}
\addbibresource{hadamard.bib}
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\begin{document}

\maketitle
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\section{Introduction}
\label{sec:introduction}

Permutation statistics are functions defined on permutations and their
generalisations. Studying the behaviour of said functions on sets of
permutations is a classical theme in algebraic and enumerative
combinatorics.
The origins of permutation statistics can be traced back to work of
Euler and MacMahon.
The past decades saw a flurry of further developments in the area; see e.g.\
\cite{BjBr/05,Stanley/97} and references therein.
Recently, Gessel and Zhuang~\cite{GZ18} developed an algebraic framework for
systematically studying so-called shuffle-compatible permutation statistics by
means of associated shuffle algebras.
In their work, quasisymmetric functions and Hadamard products of rational
generating functions played key roles.

Numerous types of zeta functions have been employed in the study of
enumerative problems surrounding algebraic structures.
L.\ Solomon~\cite{Sol77} introduced zeta functions associated
with integral representations and,
in another influential paper, Grunewald, Segal, and Smith~\cite{GSS/88}
initiated the study of zeta functions associated with (nilpotent and pro-$p$)
groups.
Following \cite{GSS/88}, a variety of methods have
been developed and applied to predict the behaviour and study symmetries
of zeta functions associated with algebraic structures, and to produce
explicit formulae.
Theoretical work of this type often employs a blend of combinatorics and
$p$-adic integration; see \cite{Vol11} for a survey.
On the practical side, a range of effective methods have
been devised and used to symbolically compute zeta functions of algebraic
structures; see \cite{Ros17/1489} and the references therein.

A common feature of zeta functions $\zeta_G(s)$ attached to algebraic
structures $G$ (e.g.\ groups) in the literature is that they often admit an
Euler factorisation $\zeta_G(s) = \prod_p \zeta_{G,p}(s)$ into so-called local
factors indexed by primes $p$.
Deep results from $p$-adic integration often guarantee that these local
factors are rational in $p^{-s}$, i.e.\ of the form $\zeta_{G,p}(s) =
W_p(p^{-s})$ for some $W_p(Y) \in \QQ(Y)$.
A key theme is then to study how the $W_p(Y)$ vary with the
prime $p$.
In a surprising number of cases of interest, deep \emph{uniformity results}
ensure the existence of a \emph{single} bivariate rational function $W(X,Y)$ such that $\zeta_{G,p}(s) =
W(p,p^{-s})$ for all primes $p$ (perhaps excluding a finite number of exceptions).
In such situations, understanding our zeta function is tantamount
to understanding $W(X,Y)$.

In this context, permutation statistics (and, more generally, combinatorial
objects) have recently found spectacular applications, in
particular when it comes to describing the numerators of the rational
functions $W(X,Y)$ from above; see, for instance,
\cite{BGS/21,CarnevaleVoll/18,CSV/19,CSV_fpsac,CSV/18,StasinskiVoll/14}.
Conversely, the need for combinatorial descriptions of such zeta functions gave
rise to new directions in the study of permutation statistics and,
more generally, combinatorial objects; see, e.g.~\cite{BC1,Carnevale/17,
  CT,Denert/90,FZ/90,DMS/23,StasinskiVoll/13,}.

In the spirit of this line of research, in the present work we relate permutation
statistics and \emph{ask zeta functions}.
Introduced in \cite{Rossmann/18} and developed further in
\cite{ask2,cico,board},
ask zeta functions are generating functions encoding average
sizes of kernels in suitable modules of matrices.
One motivation for studying these functions comes from group theory.
Indeed, for groups with a sufficiently powerful Lie theory, the enumeration
of linear orbits and conjugacy classes boils down to determining average sizes
of kernels within matrix
Lie algebras---this is essentially the orbit-counting lemma.

Amidst a plethora of algebraically-defined zeta functions,
ask zeta functions stand out as particularly amenable to combinatorial
methods.
Indeed, natural operations at the level of the modules (or groups)
often translate into natural operations of corresponding rational
generating functions.
In particular, ask zeta functions of direct sums of modules are Hadamard
products of the ask zeta function of the summands.

In this extended abstract (and forthcoming preprint \cite{upcoming}) we answer
some of the questions from \cite{cico} and give a constructive
algorithm (based on \emph{coloured shuffle compatibility} of
permutation statistics) to compute Hadamard products of certain ask
zeta functions.
Our results have corollaries pertaining to generating
functions enumerating orbits of finite direct products of groups within
various infinite families.

\section{Hadamard products and coloured configurations}
\label{elementary}

In this section, we provide a self-contained account of our main result
pertaining to Hadamard products of suitable rational generating functions.
Its proof relies on the coloured shuffle compatibility of certain permutation
statistics and the structure of associated coloured shuffle algebras.
We will describe the latter in Section~\ref{s:coloured}.

\vspace*{-1em}
\paragraph{Coloured permutations and descents.}
We consider coloured permutations
with symbols taken from the poset $\Sigma = \{ 1 < 2 < \dotsc\}$ and colours
taken from $\Gamma = \{ 0 > 1 > 2 > \dotsb\}$.
Let $\bm a = \sigma^\gamma = \sigma_1^{\gamma_1} \dotsb
\sigma_n^{\gamma_n}$ be a coloured permutation.
We write $\abs{\bm a} = n$ for the length of $\bm a$.
We further write $\symbols(\bm a) = \{ \sigma_1,\dotsc,\sigma_n\}$,
$\palette(\bm a) = \{ \gamma_1,\dotsc,\gamma_n\}$,
and
$\palette^*(\bm a) = \palette(\bm a) \setminus \{0\}$.
On the set of $\Gamma$-coloured positive integers, consider the total order
$$\dotsb <1^{1}<2^{1}<\dotsb<1^0<2^0<\dots;$$
that is, $\sigma_1^{\gamma_1}<\sigma_2^{\gamma_2}$ if and only if $\gamma_1=\gamma_2$ and
$\sigma_1<\sigma_2$, or if $\gamma_1>\gamma_2$ in $\ZZ$ (equivalently:
$\gamma_1 < \gamma_2$ in $\Gamma$).
This is the usual \emph{colour order}, corresponding to the left lexicographic
order on $\Gamma\times \Sigma$.
The \emph{descent set} of $\bm a$ as above
consists of all $i \in [n-1]$ such that $\sigma_i^{\gamma_i} >
\sigma_{i+1}^{\gamma_{i+1}}$ together with $0$ whenever $\gamma_1 \neq 0$.
The \emph{descent number} and \emph{comajor index} are defined as always as
functions of the descent set: $\des(\bm a)=\abs{\Des(\bm a)}$ and
$\comaj(\bm a)=\sum_{i\in\Des(\bm a)}(n-i)$.

\vspace*{-1em}
\paragraph{Coloured configurations.}
Let $\cA$ be the set of all coloured permutations, and
let $\ZZ\cA$ be the free abelian group with basis $\cA$.
We call elements of $\ZZ\cA$ \emph{coloured configurations}.
These elements are of the form
$f = \sum_{\bm a\in \cA} f_{\bm a} \bm a$, where each $f_{\bm a}$ belongs to
$\ZZ$ and almost all $f_{\bm a}$ are zero.
Write $\supp(f) = \{ \bm a\in \cA : f_{\bm a}\neq 0\}$, $\symbols(f) =
\bigcup_{\bm a\in \supp(f)} \symbols(\bm a)$, and $\palette^*(f) =
\bigcup_{\bm a \in \supp(f)} \palette^*(\bm a)$.
We call $f,g\in \ZZ\cA$ \emph{strongly disjoint} if $\symbols(f) \cap
\symbols(g) = \varnothing = \palette^*(f) \cap \palette^*(g)$.
For $\bm a,\bm b\in \cA$, let $\bm a\shuffle \bm b\in \ZZ\cA$ be the sum over
all shuffles of $\bm a$ and $\bm b$.
We extend $\shuffle$ to a bi-additive product on $\ZZ\cA$.

\vspace*{-1em}
\paragraph{Labels.}
Let $U = \{ \pm X^{k} : k\in \ZZ\}$, viewed as a subgroup of the multiplicative group
of the field $\QQ(X)$.
For $\alpha\colon \Gamma\to U$, write $\supp(\alpha) = \{ c\in \Gamma :
\alpha(c)\neq 1\}$ and, for $\bm a = \sigma_1^{\gamma_1} \dotsb
\sigma_n^{\gamma_n}$ as above, let $\alpha(\bm a) = \prod_{i=1}^n \alpha(\gamma_i)$.
A \emph{labelled coloured configuration} is a pair $(f,\alpha)$, where $f\in
\ZZ\cA$ and $\alpha\colon \Gamma\to U$ satisfies $\supp(\alpha) \subseteq
\palette^*(f)$.
Given labelled coloured configurations $(f,\alpha)$ and $(g,\beta)$ such that
$f$ and $g$ are strongly disjoint, the pair $(f\shuffle g, \alpha\beta)$ is a
labelled coloured configuration too.
(Here, $\alpha\beta$ denotes the pointwise product of $\alpha$ and $\beta$.)

\vspace*{-1em}
\paragraph{Equivalence.}
Let $(f,\alpha)$ be a labelled coloured configuration.
Let $\phi\colon \symbols(f) \to S$ and $\psi\colon \palette^*(f) \to P$ 
be order-preserving bijections onto finite subsets of $\Sigma$ and
$\Gamma\setminus\{0\}$, respectively.
Given $\phi$ and $\psi$, define a labelled coloured permutation
$(f',\alpha')$ as follows.
For $\bm a\in \supp(f)$, say $\bm a = \sigma_1^{\gamma_1}\dotsb
\sigma_n^{\gamma_n}$, write $\bm a' = \phi(\sigma_1)^{\psi(\gamma_1)}\dotsb
\phi(\sigma_n)^{\psi(\gamma_n)}$.
Define $f' = \sum_{\bm a\in \supp(f)}f_{\bm a}\bm a'$.
The support of $\alpha'$ is $P$ and $\alpha'(\psi(c))
= \alpha(c)$ for $c\in \palette^*(f)$.
We call $(f,\alpha)$ and each $(f',\alpha')$ (as $\phi$ and $\psi$ range
over possible choices) \emph{equivalent}, written $(f,\alpha) \asymp
(f',\alpha')$. 
This defines an equivalence relation on labelled coloured configurations.

\vspace*{-1em}
\paragraph{Rational functions.}
Given a labelled coloured configuration $(f,\alpha)$ and $\ve \in \ZZ$, we define a rational formal power series
\[
W_{f,\alpha}^\ve = W_{f,\alpha}^\ve(X,Y) = \sum_{\bm a \in \supp(f)} 
f_{\bm a} \frac{\alpha(\bm a)X^{\ve\comaj(\bm a)} Y^{\des(\bm a)} }
{(1-Y)(1-X^{\ve} Y)\dotsb(1-X^{\ve\lvert \bm a\rvert}Y)}
\in \QQ(X)[\![Y]\!].
\]
Note that, by construction, if $(f,\alpha)\asymp (f',\alpha')$, then
$W_{f,\alpha}^\ve = W_{f',\alpha'}^\ve$ for all $\ve \in \ZZ$.

\begin{ex}
  Let $f = 1^0 + 1^1$.
  Let $\alpha\colon \Gamma\to U$ with $\supp(\alpha) \subseteq\palette^*(f) =
  \{1\}$.
  Then
  \[
    W_{f,\alpha}^\ve = \frac{1 + \alpha(1) X^\ve Y}{(1-Y)(1-X^\ve Y)}.
  \]
\end{ex}

Recall that the \emph{Hadamard product} of two formal power series $A(Y) =
\sum_{k=0}^\infty a_k Y^k$ and $B(Y) = \sum_{k=0}^\infty b_k Y^k$ is
the power series $A(Y) *_Y B(Y) = \sum_{k=0}^\infty a_kb_k Y^k$.

\begin{thm}
  \label{thm:main_elementary}
  Let $(f,\alpha)$ and $(g,\beta)$ be labelled coloured configurations such that
  $f$ and $g$ are strongly disjoint.
  Then $W_{f,\alpha}^\ve *_Y W_{g,\beta}^\ve = W_{f\shuffle g, \, \alpha\beta}^\ve$ for
  each $\ve\in \ZZ$.
\end{thm}

\begin{ex}
  \label{ex:12}
  Let $f = 1^0 + 1^1$ and $g = 2^0 + 2^2$.
  Then
  \begin{align*}
    f\shuffle g & = (1^0 + 1^1)\shuffle (2^0 + 2^2) \\
    &= 1^0 \shuffle 2^0 + 1^0 \shuffle 2^2 + 1^1 \shuffle 2^0 + 1^1 \shuffle
      2^2 \\
    &= 1^0 2^0 + 2^0 1^0 + 1^0 2^2 + 2^2 1^0 + 1^1 2^0 + 2^0 1^1 + 1^1 2^2 +
      2^2 1^1.
  \end{align*}
  Let $\alpha$ and $\beta$ satisfy $\supp(\alpha) \subseteq \{1\}$ and
  $\supp(\beta) \subseteq\{2\}$.
  By Theorem~\ref{thm:main_elementary},
  \begin{align*}
    W_{f,\alpha}^\ve *_Y W_{g,\beta}^\ve
    & = 
    \frac{1+\alpha(1) X^\ve Y }{(1-Y)(1-X^\ve Y)}
    *_Y
    \frac{1+\beta(2) X^\ve Y }{(1-Y)(1-X^\ve Y)} \\
    & =
      \frac
      {1 + (1 + \alpha(1) + \beta(2))X^\ve Y +
      (\alpha(1) + \beta(2) + \alpha(1)\beta(2))X^{2\ve}Y
      + \alpha(1)\beta(2) X^{3\ve} Y^2
      }
      {(1-Y)(1-X^\ve Y)(1-X^{2\ve} Y)}
    \\ &
         = W_{f\shuffle g,\,\alpha\beta}^\ve.
  \end{align*}
\end{ex}

\noindent
Theorem~\ref{thm:main_elementary} implies, in particular, that for each
\underline{fixed} $\ve\in \ZZ$, the set
\[
  \Bigl\{ W_{f,\alpha}^\ve : (f,\alpha) \text{ is a coloured configuration}\Bigr\}
\]
is closed under Hadamard products in $Y$.
Indeed, given coloured configurations $(f,\alpha)$ and $(g,\beta)$, we can find $(g',\beta')$ such that
$f$ and $g'$ are strongly disjoint and $(g,\beta)\asymp(g',\beta')$.
In that case, $W_{f,\alpha}^\ve *_Y W_{g,\beta}^\ve = W_{f,\alpha}^\ve *_Y W_{g',\beta'}^\ve =
W_{f\shuffle g', \,\alpha\beta'}^\ve$ is computed by the preceding theorem.

In Section~\ref{s:appzeta}, we will apply Theorem~\ref{thm:main_elementary} to
provide explicit combinatorial descriptions of Hadamard products of ask,
class- and orbit-counting zeta functions.

\section{Coloured shuffle compatibility}
\label{s:coloured}

For technical reasons, in this section, we will only consider coloured
permutations with colours drawn from $\{ 0 > 1 > \dotsb > r-1\}$ (for
sufficiently large $r$).
For clarity, we occasionally refer to these as \emph{$r$-coloured permutations}.
A \emph{coloured permutation statistic} is a function $\st$ defined on the set
of coloured permutations such that given a coloured permutation $\sigma^\gamma$,
if $\pi$ is a permutation of the same length as $\sigma$ and with the same
relative order, then $\st(\sigma^\gamma) = \st(\pi^\gamma)$.
Given coloured permutation statistics $\st_1,\dotsc,\st_k$, we regard the
tuple $(\st_1,\dotsc,\st_k)$ as a coloured permutation statistic via
$(\st_1,\dotsc,\st_k)(\bm a) = (\st_1(\bm a),\dotsc,\st_k(\bm a))$.
Given a coloured permutation $\bm a = \sigma^\gamma = \sigma_1^{\gamma_1}\dotsb \sigma_n^{\gamma_n}$,
let $\col_j(\bm a) := |\{i \in [n] : \gamma_i = j\}|$.
The \emph{colour vector} of a $\bm a$ is
$\bcol(\bm a) = (\col_0(\bm a),\dotsc,\col_{r-1}(\bm a))$;
this is a weak composition of $n$.
The functions $\bcol$, $\des$, and $\comaj$ are coloured permutation statistics.

Recall from \cite{GZ18,Moustakas/th} that a (coloured) permutation
statistic $\st$ is \emph{shuffle compatible} if for coloured
permutations $\bm a$ and $\bm b$ on disjoint sets of symbols, the multiset
$\{\!\{\st(\bm c) : \bm c \in \bm a\shuffle \bm b\}\!\}$
only depends on $\st(\bm a), \st(\bm b)$ and the lengths of $\bm a$ and $\bm b$.
(Here, $\bm a\shuffle \bm b$ denotes the set of all coloured permutations obtained as
shuffles of $\bm a$ and $\bm b$.)
Generalising \cite{GZ18, Moustakas/th},
we associate a shuffle algebra $\A_{\st}^{(r)}$ over $\QQ$ to 
a shuffle-compatible coloured permutation statistic $\st$ as follows.
First, $\st$ defines an equivalence relation $\sim_{\st}$ on $r$-coloured permutations via
$\bm a\sim_{\st} \bm b$ if and only if
$\bm a$ and $\bm b$ have the same length and
$\st(\bm a)=\st(\bm b)$; we refer to this as
\emph{$\st$-equivalence}. 
We write $[\bm a]_{\st}$ to denote the $\st$-equivalence class of
$\bm a$.
As a $\QQ$-vector space $\A_{\st}^{(r)}$ has a basis given by the
$\st$-equivalence classes of $r$-coloured permutations.
The multiplication is given by linearly extending the rule
$$
[\bm a]_{\st} \, [\bm b]_{\st} = \sum_{\bm c \in \bm a
  \shuffle\bm b} [\bm c]_{\st},
$$
where $\bm a$ and $\bm b$ are $r$-coloured permutations on disjoint sets of symbols.
(Thanks to the shuffle compatibility of $\st$, this yields a well-defined
multiplication on $\A_{\st}^{(r)}$.)

The main shuffle algebra of interest to us
is the one attached to $(\des,\comaj,\bcol)$.
Let $p_0,\dotsc,p_{r-1},x,t$ be commuting variables over $\QQ$;
write $\bp = (p_0,\dotsc,p_{r-1})$.
For a ring $R$, let $R[\![t\Asterisk]\!]$ denote the ring
$R[\![t]\!]$ with multiplication given by the Hadamard product in~$t$.

\begin{thm}\label{thm:triple_compatibility}
  \quad
  \begin{enumerate}
  \item[(a)]
    The tuple of statistics $(\des,\comaj,\bcol)$ is shuffle compatible.
  \item[(b)]
The linear map on $\A_{(\des,\comaj,\bcol)}^{(r)}$ defined by 
\[
[\bm a]_{(\des,\comaj,\bcol)} \mapsto
\begin{cases}
  \frac{\bp^{\bcol(\bm a)} x^{\comaj(\bm a)} t^{\des(\bm a)+1}
  }{(1-t)(1-xt)\cdots(1-x^{\lvert \bm a\rvert}t)} z^{\lvert \bm a\rvert},
  &\text{if $\lvert \bm a\rvert \ge 1$} \\
  \frac{1}{1-t}, &\text{if $\lvert \bm a\rvert=0$}
\end{cases}
\] 
is an isomorphism from the ($r$-coloured)
shuffle algebra of $(\des,\comaj,\bcol)$ onto the subalgebra of
$\mathbb Q[p_0,p_1,\dots,p_{r-1},z,x] [\![t\Asterisk]\!]$ spanned by
%
\begin{align*}
  \left\lbrace \frac{1}{1-t}\right\rbrace \bigcup 
  \bigg\lbrace \frac{p_0^{c_0} \dotsb p_{r-1}^{c_{r-1}} \, x^k
  \,t^{j+1} }{(1-t)(1-xt)\cdots(1-x^nt)} z^n \bigg\rbrace_{n \ge 1, \,j \in
  [0,n],\, c_0, \dotsc, c_{r-1} \in [0,r-1], \, k \in \left[\binom{j+1}{2}, nj - \binom{j}{2}\right]}.
\end{align*}
  \end{enumerate}
\end{thm}

While we omit the proof of the preceding theorem here, we should like to take
this opportunity to provide a brief overview of its key steps and ingredients.
The \emph{coloured descent set} of a coloured permutation $\bm a =
\sigma_1^{\gamma_1}\dotsb \sigma_n^{\gamma_n}$ is defined as
$$\sDes(\bm a)=\Bigl\{(i,\gamma_i): i \in[n-1], \, \gamma_i\neq\gamma_{i+1} \text{ or }
(\gamma_i=\gamma_{i+1} \text{ and }\sigma_i>\sigma_{i+1})\Bigr\} \cup
\Bigl\{(n,\gamma_n)\Bigr\}.$$
This coloured permutation statistic, which was introduced by Mantaci and
Reutenauer \cite{MR/95} while studying a coloured generalisation of Solomon's
descent algebra, is shuffle compatible.
Moreover, it refines the tuple $(\des,\comaj,\bcol)$.
As a consequence, the algebra $\A_{(\des,\comaj,\bcol)}^{(r)}$ is naturally a
quotient of $\A_{\sDes}^{(r)}$.

Let $x_i^{(j)}$ for $i=1,2,\dotsc$ and $j=0,1,\dotsc,r-1$ be independent
(commuting) variables. We write $\bx^{(j)} = (x_1^{(j)}, x_2^{(j)}, \dots)$.
The \emph{coloured quasisymmetric function} attached to an $r$-coloured
permutation $\bm a = \sigma^\gamma$ of length $n$ is
% 
 \begin{equation*}
   F_{\bm a}(\bx^{(0)}, \dots, \bx^{(r-1)}) =
   \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_n\\j \in \Des^*\!(\bm a) \, \Rightarrow \, i_j < i_{j+1}}} x_{i_1}^{(\gamma_1)}x_{i_2}^{(\gamma_2)}\cdots x_{i_n}^{(\gamma_n)},
\end{equation*}
%
where $\Des^*\!(\bm a) = \Des(\bm a)\setminus \{0\}$.
This is a (homogeneous)
formal power series of degree $n$ in the variables
$\bx^{(0)}, \dots, \bx^{(r-1)}$.
These functions were first introduced in \cite{Poirier}; see also
\cite{HP/10,Moustakas/th,moustakas}.
The space $\QSym^{(r)}$ spanned by all such coloured quasisymmetric
functions forms a $\QQ$-algebra.
It turns out that $\QSym^{(r)}$ and $\A_{\sDes}^{(r)}$ are canonically
isomorphic.
Our proof of Theorem~\ref{thm:triple_compatibility} is based on a judicious
choice of specialisations of coloured quasisymmetric functions;
cf.\ \cite[\S 4.4]{Moustakas/th}.
Theorem~\ref{thm:main_elementary} is a consequence of
Theorem~\ref{thm:triple_compatibility}.

\section{Applications to zeta functions}
\label{s:appzeta}

\subsection{Ask, class-counting, and orbit-counting zeta functions}
\label{ss:ask}

The main purpose of the present section is to recall four families of
zeta functions associated with algebraic structures
(Examples~\ref{ex:ask_mat}--\ref{ex:graphs}).
These will feature in our applications of Theorem~\ref{thm:main_elementary} in
Section~\ref{ss:app}.
For further details, see \cite{Rossmann/18,ask2}.
Rings will be assumed to be commutative and unital.
In order to maintain consistency with the literature, we
regard $d\times e$ matrices over a ring $R$ as homomorphisms $R^d \to
R^e$ acting by right multiplication.

\vspace*{-1em}
\paragraph{Global ask zeta functions.}
Given a module $M\subseteq \Mat_{d\times e}(\ZZ)$ of integral matrices, for
each $n\ge 1$, let $M_n\subseteq \Mat_{d \times e}(\ZZ/n\ZZ)$ denote the
reduction of $M$ modulo $n$.
The \emph{(global) ask zeta function} of $M$ is the Dirichlet series
$\zeta_M^{\ask}(s) = \sum_{n=1}^\infty a_n(M) n^{-s}$, where $a_n(M) \in \QQ$ denotes the
average size of the kernel of matrices in $M_n$.
By the Chinese remainder theorem, 
$\zeta^{\ask}_M(s) = \prod_p \zeta^{\ask}_{M_p}(s)$ (Euler product), where
the product is taken over all primes $p$ and the local factor at $p$ is
given by $\zeta^{\ask}_{M_p}(s) = \sum_{k=0}^\infty a_{p^k}(M) p^{-ks}$,
a power series in $p^{-s}$.
Drawing upon deep results from $p$-adic integration and the theory of zeta
functions of algebraic structures,
it is known that each $\zeta^{\ask}_{M_p}(s)$ is rational in $p^{-s}$.

\vspace*{-1em}
\paragraph{Local ask zeta functions.}
It is often advantageous to bypass global structures altogether and directly
study variants of the local factors from above.
Let $\mathfrak O$ be a compact discrete valuation ring.
Let $\mathfrak P$ be the maximal ideal of $\mathfrak O$ and let $q$
denote the size of the residue field $\mathfrak O/\mathfrak P$.
Such rings $\mathfrak O$ are precisely the valuation rings of non-Archimedean
local fields.
Examples include the $p$-adic integers $\ZZ_p$ (in which case
$\mathfrak O/\mathfrak P\cong \FF_p$) and the ring $\FF_q[\![z]\!]$ of formal
power series over $\FF_q$ (in which case $\mathfrak P = z \FF_q[\![z]\!]$).

Given a module of matrices $M\subseteq \Mat_{d\times e}(\mathfrak O)$,
its associated \emph{(local) ask zeta function} is the formal power series
$\Zeta^{\ask}_M(Y) = \sum_{k=0}^\infty \alpha_k(M) Y^k$, where $\alpha_k(M)$
denotes the average size of the kernels within the reduction of $M$ modulo
$\mathfrak P^k$.

\begin{ex}
  \label{ex:ask_mat}
  $\Zeta^{\ask}_{\Mat_{d\times e}(\mathfrak O)}(Y) = \frac{1-
  q^{-e}Y}{(1-Y)(1-q^{d-e}Y)}$; see \cite[Prop.~1.5]{Rossmann/18}.
\end{ex}

\begin{ex}
  \label{ex:ask_so}
  Let $\mathfrak O$ have characteristic distinct from $2$.
  Let $\mathfrak{so}_d(\mathfrak O)$ be the module of antisymmetric $d\times d$
  matrices over $\mathfrak O$.
  By \cite[Prop.~5.11]{Rossmann/18},
  $\Zeta^{\ask}_{\mathfrak{so}_d(\mathfrak O)}(Y) = \frac{1-q^{1-d} Y}{(1-Y)(1-qY)}$.
\end{ex}

\begin{ex}
  \label{ex:ask_n}
  Let $\mathfrak n_d(\mathfrak O)$ be the module of strictly upper triangular
  $d\times d$ matrices over $\mathfrak O$.
  By \cite[Prop.~5.15(i)]{Rossmann/18},
  $\Zeta^{\ask}_{\mathfrak n_d(\mathfrak O)}(Y) = \frac{(1-Y)^{d-1}}{(1-qY)^d}$.
\end{ex}

\vspace*{-1em}
\paragraph{Class- and orbit-counting zeta functions.}
Let $\mathfrak O$ be a compact discrete valuation ring as above.
Let $\mathsf G$ be a linear group scheme over $\mathfrak O$, with a given embedding
into $d\times d$ matrices.
The \emph{orbit-counting zeta function} of $\mathsf G$ is the generating function
$\Zeta^{\oc}_{\mathsf G}(Y) = \sum_{k=0}^\infty b_k(\mathsf G) Y^k$,
where $b_k(\mathsf G)$ denotes the number of orbits of the (finite) matrix
group $\mathsf G(\mathfrak O/\mathfrak P^k)$ on its natural module $(\mathfrak
O/\mathfrak P^k)^d$.
The \emph{class-counting zeta function} of $\mathsf G$ is the generating
function $\Zeta^{\concnt}_{\mathsf G}(Y) = \sum_{k=0}^\infty c_k(\mathsf G) Y^k$,
where $c_k(\mathsf G)$ denotes
the number of conjugacy classes of $\mathsf G(\mathfrak O/\mathfrak P^k)$.
Class-counting zeta functions go back to work of
du~Sautoy~\cite{dS05}.
As shown in \cite{Rossmann/18,ask2}, subject to restrictions on the residue
field size $q$ of $\mathfrak O$, class- and orbit-counting zeta functions
of $\mathsf G$ are instances of ask zeta functions associated with modules of matrices
over $\mathfrak O$.
When passing between ask and class-counting zeta functions, one
often needs to apply a transformation $Y\gets q^m Y$ for a suitable integer
$m$;
see below for an example and cf.\ Lemma~\ref{lem:shift}.

\begin{ex}
  \label{ex:cc_F2d}
  Suppose that the residue field size $q$ of $\mathfrak O$ is odd.
  By exponentiation, the free class-$2$-nilpotent Lie algebra on $d$
  generators over $\mathfrak O$ gives rise to a group scheme $\mathsf F_{2,d}$ over
  $\mathfrak O$.
  We may view $\mathsf F_{2,d}$ as an analogue of the free
  class-$2$-nilpotent group on $d$ generators.
  Lins \cite[Cor.\ 1.5]{Lins/20} showed that
  {\small $$\Zeta^{\cc}_{\mathsf F_{2,d}}(Y) = \frac{1-q^{\binom{d-1} 2}Y}{\Bigl(1 - q^{\binom d
      2}Y\Bigr)\Bigl(1- q^{\binom d 2 + 1}Y\Bigr)}.$$}
  Looking back at Example~\ref{ex:ask_so}, we observe that
  $\Zeta^{\cc}_{\mathsf F_{2,d}}(Y) = \Zeta^{\ask}_{\mathfrak{so}_d(\mathfrak
    O)}(q^{\binom d 2}Y)$;
  this is no coincidence, see \cite[Ex.~7.3]{ask2}.
\end{ex}

\begin{ex}
  \label{ex:oc_Ud}
  Let $\mathsf U_d$ be the group scheme of upper unitriangular $d\times d$ matrices over
  $\mathfrak O$.
  Suppose that $\gcd(q, (d-1)!) = 1$.
  By \cite[Thm~1.7]{Rossmann/18} (cf.~\cite[Prop.\ 4.12]{board})
  and Example~\ref{ex:ask_n}, we have
  $\Zeta^{\oc}_{\mathsf U_d}(Y) = \frac{(1-Y)^{d-1}}{(1-qY)^{d}}$.
\end{ex}

\vspace*{-1em}
\paragraph{Graphs and graphical groups.}
Given a (finite, simple) graph $\Gamma$ with distinct vertices
$v_1,\dotsc,v_n$ and $m$ edges, let $M_\Gamma$ be the module of antisymmetric $n\times n$
matrices $A = [a_{ij}]$ such that $a_{ij} = 0$ whenever $v_i$ and $v_j$ are
non-adjacent.
We write $\Zeta^{\ask}_{\Gamma}(Y)$ for $\Zeta^{\ask}_{M(\Gamma)}(Y)$.
As shown in \cite[Thm~A]{cico}, $\Zeta^{\ask}_{\Gamma}(Y)$ is a rational
function in $q$ and $Y$.
In \cite[\S 3.4]{cico}, the graphical group scheme $\mathsf G_\Gamma$
associated with $\Gamma$ is constructed; for an alternative but equivalent
construction, see \cite[\S 1.1]{higman}.
By \cite[Prop.\ 3.9]{cico}, $\Zeta^{\cc}_{\mathsf G_\Gamma}(Y) =
\Zeta^{\ask}_{\Gamma}(q^m Y)$.
Given graphs $\Gamma_1$ and~$\Gamma_2$, let $\Gamma_1 \vee \Gamma_2$ denote
their join, obtained from the disjoint union of $\Gamma_1$ and $\Gamma_2$ by
adding edges connecting each vertex of $\Gamma_1$ to each vertex of $\Gamma_2$.
Let $\mathrm K_n$ (resp.\ $\Delta_n$) denote the complete (resp.\ edgeless)
graph on $n$ vertices. 

\begin{ex}
  \label{ex:graphs}
  Consider $\Gamma = \Delta_n \vee \mathrm K_{n+1}$.
  Then $\Gamma$ has $3\binom{n+1} 2$ edges.
  It follows from \cite[Thm~8.18]{cico} that
  $$
  \Zeta^{\ask}_{\Gamma}(Y) = \frac{(1-q^{-n}Y)(1-q^{-n-1}Y)}{(1-q^{-1}Y)(1-Y)(1-qY)}.
  $$
\end{ex}

\vspace*{-1em}
\paragraph{Hadamard products and zeta functions.}

Let $\mathfrak O$ be as above.
Elaborating further on what we wrote in the introduction,
modules of matrices, (linear) group schemes, and graphs all admit natural
operations which correspond to taking Hadamard products of zeta functions.
In detail, given modules $M \subseteq \Mat_{d\times e}(\mathfrak O)$ and
$M'\subseteq \Mat_{d'\times e'}(\mathfrak O)$, we regard $M\oplus M'$ as a
submodule of $\Mat_{(d+d')\times (e+e')}(\mathfrak O)$, embedded diagonally.
Then $\Zeta^{\ask}_{M\oplus M'}(Y) = \Zeta^{\ask}_{M}(Y) *_Y
\Zeta^{\ask}_{M'}(Y)$.
Similarly, given (linear) group schemes $\mathsf G$ and $\mathsf G'$ over $\mathfrak O$,
we obtain $\Zeta^{\cc}_{\mathsf G\times \mathsf G'}(Y) = \Zeta^{\cc}_{\mathsf G}(Y) *_Y
\Zeta^{\cc}_{\mathsf G'}(Y)$.
Finally, given graphs $\Gamma$ and $\Gamma'$, let $\Gamma \oplus \Gamma'$
denote their disjoint union.
Then
$\Zeta^{\ask}_{\Gamma \oplus \Gamma'}(Y) = \Zeta^{\ask}_{\Gamma}(Y) *_Y
\Zeta^{\ask}_{\Gamma'}(Y)$; moreover, $\mathsf G_{\Gamma\oplus \Gamma'} \cong
\mathsf G_{\Gamma} \times \mathsf G_{\Gamma'}$.

\subsection{Applications}
\label{ss:app}

It turns out that each zeta function from
Examples~\ref{ex:ask_mat}--\ref{ex:graphs}
can be expressed in terms of the rational functions $W_{f,\alpha}^e(X,Y)$
attached to labelled coloured configurations as in Section~\ref{elementary}.
Omitting proofs, details of this are recorded in Table~\ref{tab}.
This table is to be read as follows: for each row and each compact discrete
valuation ring $\mathfrak O$ with residue field size~$q$ (possibly subject to
further conditions on $\mathfrak O$ as in the examples above), the zeta function
$\Zeta(Y)$ indicated in the leftmost column is obtained from the rational
function in the rightmost column via $\Zeta(Y) = W_{f,\alpha}^\ve(q,u(q) Y)$.
In Table~\ref{tab}, we write $\underline n$ for the sum of all $2^n$ coloured
permutations of the form $1^{\nu_1} \dotsb n^{\nu_n}$ with $\nu_i\in \{0,i\}$.
A ``\dup{}'' indicates that an entry coincides with the one immediately
above it.
We note that Table~\ref{tab} does not constitute an exhaustive list of zeta
functions expressible in terms of coloured configurations; we refer to our
upcoming paper \cite{upcoming} for further examples and applications.

\begin{table}[h]
  \centering
  \begin{tabular}{|c|c|c|c|c|c|}
    \hline
    {Zeta function} & $f$ & $\alpha$ & $\ve$ & $u(X)$ & $W_{f,\alpha}^\ve(X,Y)$ \\ \hline
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    $\Zeta^{\ask}_{\Mat_{d\times e}(\mathfrak O)}(Y)$ & $\underline 1$ & $1 \gets -X^{-d}$ & \small $d - e$ & $1$ & $\frac{1-X^{-e}Y}{(1-Y)(1-X^{d-e}Y)}$ \\ \hline
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \small $\Zeta^{\ask}_{\mathfrak{so}_d(\mathfrak O)}(Y)$, \,$\Zeta^{\ask}_{\Mat_{d\times(d-1)}(\mathfrak O)}(Y)$ & \dup & \dup & $1$ & \dup & $\frac{1-X^{1-d}Y}{(1-Y)(1-XY)}$ \\ \hline
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    $\Zeta^{\cc}_{\mathsf F_{2,d}}(Y)$ & \dup & \dup & \dup & $X^{\binom d 2}$ & \dup \\ \hline
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    $\Zeta^{\ask}_{\Delta_n\vee \mathrm K_{n+1}}(Y)$ & \underline 2 & \small $1,2\gets -X^{-n-1}$ & $1$ & $X^{-1}$ & \small $\frac{(1-X^{1-n}Y)(1-X^{-n}Y)}{(1-Y)(1-XY)(1-X^2Y)}$ \\ \hline
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    $\Zeta^{\cc}_{\mathsf G_{\Delta_n\vee \mathrm K_{n+1}}}(Y)$ & \dup & \dup & \dup & \small $X^{3\binom{n+1} 2-1}$ & \dup \\ \hline
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    $\Zeta^{\oc}_{\mathsf U_{d+1}}(Y)$ & \small $\underline{d}$ & \small $1,\dotsc,d\gets -X^{-1}$ & 0 & $X$ & $\frac{(1-X^{-1}Y)^{d}}{(1-Y)^{d+1}}$ \\\hline
  \end{tabular}
  \caption{Examples of zeta functions from labelled coloured configurations}
  \label{tab}
\end{table}

We now explain how, subject to a compatibility condition,
Theorem~\ref{thm:main_elementary} can be used to explicitly compute Hadamard
products of the zeta functions in Table~\ref{tab}.
As explained in Section~\ref{ss:ask}, we can interpret such Hadamard products as zeta
functions associated with
 ``products''
of the objects under
consideration.
We first record an elementary fact.

\begin{lem}
  \label{lem:shift}
  Let $R$ be a commutative ring.
  Let $A(Y) = \sum_{k=0}^\infty a_k Y^k$ and $B(Y) = \sum_{k=0}^\infty b_k
  Y^k$ be formal power series over $R$.
  Let $u,v\in R$.
  Then $A(uY) *_Y B(vY) = (A *_Y B)(uvY)$. \qed
\end{lem}

The compatibility condition that we alluded to above is that we require
entries in the $\ve$-column of Table~\ref{tab} to agree for us to compute
associated Hadamard products via Theorem~\ref{thm:main_elementary} and
Lemma~\ref{lem:shift}.
Thus, suppose that $(f,\alpha)$ and $(g,\beta)$ are coloured configurations
and let $u(X)$ and $v(X)$ each be of the form $\pm X^\ell$ for $\ell\in \ZZ$.
Then
$$W_{f,\alpha}^\ve\bigl(X,\, u(X)Y\bigr) *_Y W_{g,\beta}^\ve\bigl(X,\, v(X) Y\bigr)
= (W_{f,\alpha}^\ve *_Y W_{g,\beta}^\ve)\bigl(X,\,u(X)v(X)Y\bigr).$$

As explained in Section~\ref{elementary}, by passing to equivalent labelled coloured
configurations, we may assume that $f$ and $g$ are strongly disjoint.
Theorem~\ref{thm:main_elementary} then allows us to explicitly compute
$W_{f,\alpha}^\ve *_Y W_{g,\beta}^\ve$. (Here it is crucial that a common
value of $\ve$ is used in both factors.)
All that remains to obtain our zeta function is to apply the
specialisation $X\gets q$.

To illustrate the scope of our method by means of, say, a group-theoretic
application, first note that for \textit{specific} choices of $d_1,\dotsc,d_r$,
a finite computation (using the algorithm in \cite[\S 6]{cico}, implemented in
the software package \textsf{Zeta}~\cite{Zeta}) can be used to determine
$\Zeta^{\concnt}_{\mathsf F_{2,d_1}\times\dotsb\times \mathsf F_{2,d_r}}(Y)$.
Our results here go substantially further.
Indeed, for any fixed $r$, an explicit finite computation produces a
single formula for $\Zeta^{\concnt}_{\mathsf F_{2,d_1}\times\dotsb\times
  \mathsf F_{2,d_r}}(Y) = 
\Zeta^{\concnt}_{\mathsf F_{2,d_1}}(Y) *_Y \dotsb *_Y \Zeta^{\concnt}_{\mathsf F_{2,d_r}}(Y)$
as a \emph{symbolic expression} involving variables $d_1,\dotsc,d_r$.

\begin{ex}
  By combining Example~\ref{ex:12} and Example~\ref{ex:cc_F2d},
  we find that $\Zeta^{\concnt}_{\mathsf F_{2,d}\times \mathsf F_{2,d'}}(Y) =
  W\bigl(q,q^{\binom d 2 + \binom{d'}2}Y\bigr)$, where
  $$
  W(X,Y) = \frac{1 + (1 - X^{-d} - X^{-d'})XY + (X^{-d-d'} - X^{-d} -
    X^{-d'})X^2Y + X^{-d-d'}X^3Y^2}{(1-Y)(1-XY)(1-X^2Y)}.
  $$
\end{ex}

In the same spirit, using the data from Table~\ref{tab}, we can e.g.\
symbolically compute the orbit-counting zeta function of $\mathsf
U_{d_1}\times \dotsb \times \mathsf U_{d_r}$ and the ask zeta function
of $\Mat_{d_1\times e_1}(\mathfrak O)\oplus \dotsb \oplus \Mat_{d_r\times
  e_r}(\mathfrak O)$ when $d_1 - e_1 = \dotsb = d_r - e_r$,
all for fixed $r$ but symbolic variables $d_1,\dotsc,d_r$ and $e_1,\dotsc,e_r$.
In particular, the latter case provides an explicit description of the ask zeta
function of $\Mat_{d_1}(\mathfrak O) \oplus \dotsb \oplus \Mat_{d_r}(\mathfrak
O)$ in terms of coloured permutation statistics;
cf.\ \cite[Question~10.4(a)--(b)]{cico}.
Such a description was previously only known for $d_1=\dotsb=d_r$; see
\cite[Prop.\ 10.3]{cico}.

We conclude by noting that, to the best of our
knowledge,
the previously mentioned result \cite[Prop.\ 10.3]{cico} and its close
relative \cite[Cor.\ 5.17]{Rossmann/18} (both pertaining to direct powers of
$\Mat_d(\mathfrak O)$) were the only examples of Hadamard products of ask (as
well as class- and orbit-counting) zeta functions expressed in terms of
(coloured) permutation statistics.
In the aforementioned results in the literature, the proofs rely on work of
Brenti~\cite{Bre94}.
These proofs are based on the coincidence of the rational generating
functions in question with those attached to so-called $q$-Eulerian
polynomials of signed permutations.
This preceded more recent machinery surrounding shuffle
compatibility.
This earlier work is now explained as part of the framework presented here.

\acknowledgements{AC and TR gratefully acknowledge interesting discussions with Christopher Voll.}

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