Multivariate Dirichlet Moments and a Polychromatic ESF Multivariate Dirichlet
Moments and
a Polychromatic Ewens Sampling FormulaThis research was funded by the Austrian
Science Fund (FWF) ESPRIT 208. For the purpose of open access, the authors
have applied a CC BY public copyright licence to any Author Accepted
Manuscript version arising from this submission. F.Q. gratefully acknowledges
support by the Austrian Science Fund (FWF), Project SFB F65. The authors are
grateful to Professor Nathanaël Berestycki for several helpful suggestions,
and to Nicola Battisti and Dr. Elizabeth Hollwey for enlightening discussions
on DNA-methylation. Lorenzo Dello Schiavo 111Institute of Science and
Technology Austria, Austria<EMAIL_ADDRESS>and Filippo
Quattrocchi 222Institute of Science and Technology Austria, Austria.
<EMAIL_ADDRESS>Dirichlet distribution; Ewens sampling formula;
Hoppe urn model; colored partitions 60C05 (Primary), 60J10 0 0 0 0 We present
an elementary non-recursive formula for the multivariate moments of the
Dirichlet distribution on the standard simplex, in terms of the pattern
inventory of the moments’ exponents. We obtain analog formulas for the
multivariate moments of the Dirichlet–Ferguson and Gamma measures. We further
introduce a polychromatic analogue of Ewens sampling formula on colored
integer partitions, discuss its relation with suitable extensions of Hoppe’s
urn model and of the Chinese restaurant process, and prove that it satisfies
an adapted notion of consistency in the sense of Kingman.

##  1 Introduction

We present extensions of some celebrated models of random integer partitions,
to the case when such partitions are decorated by a subordinate specification,
for simplicity described as a categorically distributed coloring. The _fil
rouge_ of our presentation is an algebraic approach to the count of integer
partitions, which we draw from well-known connections among the _Dirichlet
distribution_ , _Ewens sampling formula_ (ESF), _Hoppe’s urn model_ , the
_Chinese restaurant process_ (CRP), etc.

Our starting point is the observation that univariate moments of the Dirichlet
distribution are the generating functions of the (standard, ‘monocromatic’)
ESF (cf. (3.3) below). Here, our goal is to describe the relation between
_multi_ variate moments of the Dirichlet distribution and a ‘ _poly_
chromatic’ ESF on colored partitions. A systematic treatment of the arising
‘colored partition structure’, including a representation theorem in the sense
of Kingman [17], will be the subject of future work.

Denote by $\Gamma$ the _Euler Gamma function_ , by
$\left\langle\alpha\right\rangle_{k}\operatorname{\coloneqq}\Gamma(\alpha+k)/\Gamma(\alpha)$
the _Pochhammer symbol_ of $\alpha>0$, and by
$\mathrm{B}(x_{1},\dotsc,x_{k})\operatorname{\coloneqq}\Gamma(x_{1})\cdots\Gamma(x_{k})/\Gamma(x_{1}+\cdots+x_{k})$
the _multivariate Euler Beta function_. For $k\geq 1$ further let
$\Delta^{k-1}$ be the _standard simplex_ (3.1). For
$\boldsymbol{\alpha}\in{\mathbb{R}}_{+}^{k}$, the _Dirichlet distribution_
$D_{\boldsymbol{\alpha}}$ is the probability measure with density

$\frac{\operatorname{\mathbf{1}}_{\Delta^{k-1}}(x_{1},\dotsc,x_{k})}{\mathrm{B}(\alpha_{1},\dotsc,\alpha_{k})}\,x_{1}^{\alpha_{1}-1}\cdots
x_{k}^{\alpha_{k}-1}$

w.r.t. the standard Lebesgue measure on the hyperplane of equation
$x_{1}+\cdots+x_{k}=1$.

##### Moments of Dirichlet measures

To find useful representations for the moments of $D_{\boldsymbol{\alpha}}$ is
a difficult problem, of which we present a brief historical account in §3.1.
As a first main result, we provide a simple, elementary, closed formula for
all multivariate moments of $D_{\boldsymbol{\alpha}}$. Precisely, fix integers
$q\in{\mathbb{N}}_{1}$ and
$\mathbf{n}\operatorname{\coloneqq}\left(n_{1},\dotsc,n_{q}\right)\in{\mathbb{N}}_{1}^{q}$,
and let $\mathscr{Z}_{\mathbf{n}}$ be the _pattern inventory_ (2.6) of
$\mathbf{n}$, also see (2.9).

######  Theorem 1 (see Thm. 3.1).

For every $\mathbf{s}_{1},\dotsc,\mathbf{s}_{q}\in{\mathbb{C}}^{k}$ and
$\boldsymbol{\alpha}\in{\mathbb{R}}^{k}_{+}$,

$\int_{\Delta^{k-1}}\prod_{j=1}^{q}(\mathbf{s}_{j}\cdot\mathbf{y})^{n_{j}}\mathop{}\\!\mathrm{d}D_{\boldsymbol{\alpha}}(\mathbf{y})=\frac{n_{1}!\cdots
n_{q}!}{\left\langle\alpha_{1}+\cdots+\alpha_{k}\right\rangle_{n_{1}+\cdots+n_{q}}}\,\mathscr{Z}_{\mathbf{n}}[\mathbf{s}_{1},\dotsc,\mathbf{s}_{q};\boldsymbol{\alpha}]\,\mathrm{.}$
(1.1)

By ‘simple’ we mean that our formula is not further simplifiable in terms of
actions of the symmetric groups
$\mathfrak{S}_{n_{1}},\dotsc,\mathfrak{S}_{n_{q}}$, by ‘elementary’ that it is
expressed only in terms of elementary functions, and by ‘closed’ that it is
both non-recursive and non-iterative.

##### Ewens Sampling Formula

For a permutation $\pi$ in the symmetric group $\mathfrak{S}_{n}$, denote by
$r\operatorname{\coloneqq}r(\pi)$ the total number of its cycles (including
fixed points). Let $\theta>0$ and recall that a probability distribution on
$\mathfrak{S}_{n}$ is _$\theta$ -biased_ if its value on each $\pi$ is
proportional to $\theta^{r}$. The _Ewens Sampling Formula_ (ESF) with
parameter $\theta$ is the probability distribution

$E_{\theta}(\boldsymbol{\lambda})\operatorname{\coloneqq}\frac{n!}{\left\langle\theta\right\rangle_{n}}\prod_{i=1}^{n}\frac{\theta^{\lambda_{i}}}{i^{\lambda_{i}}\lambda_{i}!}\,\mathrm{,}\;\,\qquad\boldsymbol{\lambda}\operatorname{\coloneqq}\left(\lambda_{1},\dotsc,\lambda_{n}\right)\,\mathrm{,}\;\,$

on the set of integer partitions $\boldsymbol{\lambda}$ of $n$, i.e.
satisfying $\sum_{i}i\lambda_{i}=n$. It is the probability that a
$\theta$-biased permutation has given cycle structure $\boldsymbol{\lambda}$,
i.e. with $\lambda_{1}$ fixed points, $\lambda_{2}$ transpositions,
$\lambda_{3}$ $3$-cycles, etc. In particular, the distribution $E_{1}$
describes the frequency of a permutation in $\mathfrak{S}_{n}$ with a given
cycle structure.

We refer the reader to the recent surveys [3, 28] and references therein for a
complete account of the history and importance of the ESF throughout
mathematics and beyond.

##### A Polychromatic ESF

The proof of Theorem Theorem 1 will partly consist in counting the cardinality
of the orbits of a certain group action with homogeneous space the symmetric
group $\mathfrak{S}_{n_{1}+\cdots+n_{q}}$. As a byproduct we derive a
_polychromatic ESF_ which we now describe.

For positive integers $q$ and
$\mathbf{n}\operatorname{\coloneqq}\left(n_{1},\dotsc,n_{q}\right)$ we set
$n\operatorname{\coloneqq}n_{1}+\cdots+n_{q}$ and consider the set
$[n]\operatorname{\coloneqq}\left\\{1,\dotsc,n\right\\}$. We interpret
$[q]\operatorname{\coloneqq}\left\\{1,\dotsc,q\right\\}$ as a set of colors
—or, more generally, of categories— and assign color $c_{1}$ to $n_{1}$
elements of $[n]$, color $c_{2}$ to $n_{2}$ elements, and so on, in a fixed
deterministic way. Taking into account the coloring of the elements in $[n]$,
one may ask for the following refinement of the standard ESF.

######  Question 1.

What is the probability that a $\theta$-biased random permutation
$\pi\in\mathfrak{S}_{n}$, has a given cycle structure and each orbit of $\pi$
has a given number of elements of color $c_{j}$, $j\in[q]$?

In order to answer Question 1, it is convenient to encode both the cycle
structure of $\pi$ and the number of $c_{j}$-colored elements in each cycle
(orbit) of $\pi$ into a multiset, namely a _$q$ -colored partition_ which we
now describe; also see Dfn. 2.2 below. Suppose that
$\pi=\kappa_{1}\cdots\kappa_{r}$ is a permutation with cycles $\kappa_{i}$,
including (!) fixed points. To each cycle $\kappa=(y_{1}\cdots y_{m})$ of
$\pi$ we associate its _color count_ , i.e. the vector
$\mathbf{a}=(a_{1},\dotsc,a_{q})$ where $a_{j}$ is the number of elements of
color $c_{j}$ in $\left\\{y_{1},…,y_{m}\right\\}\subset[n]$. The colored
partition associated to $\pi$ is the function $A$ assigning to each fixed
$\mathbf{a}$ the number of cycles $\kappa$ of $\pi$ with color count
$\mathbf{a}$. We say that $\pi$ _has (cycle structure and) coloring $A$_. As
it turns out, the number of permutations with given coloring $A$ is the
multinomial coefficient (2.2) of $A$.

Now, let $\theta>0$ be a rate parameter, and $\mathbf{p}\in\Delta^{q-1}$ be
the parameter of a categorical distribution on $[q]$. We define a probability
measure $E_{\theta,\mathbf{p}}^{n}$ (Dfn. 4.1) on the set of all $q$-colored
partitions of $n$, the properties of which we collect hereafter.

######  Theorem 2 (Polychromatic ESF).

For every $\theta>0$ and every $\mathbf{p}\in\Delta^{q-1}$,

  1. $(i)$

when $q=1$, hence $\mathbf{p}=p=1$, $E_{\theta,1}^{n}$ is the Ewens
distribution $E_{\theta}$ on partitions of $n$;

  2. $(ii)$

(Prop. 4.4) conditioning $E_{\theta,\mathbf{p}}^{n}$ on a $q$-colored
partition $A$ coloring $\mathbf{n}$ gives the probability that a
$\theta$-biased random permutation $\pi$ has cycle structure and coloring $A$;
(This answers Question 1.)

  3. $(iii)$

(Prop. 4.7) $E_{\theta,\mathbf{p}}^{n}$ is the marginal distribution at time
$n$ of the _polychromatic Hoppe urn model_ described in §4.1 and of the
extension of the CRP described below;

  4. $(iv)$

(Thm. 4.10) the family $E_{\theta,\mathbf{p}}^{n}$, $n\in{\mathbb{N}}_{1}$, is
_consistent_ in a suitable sense extending the notion of Kingman’s consistency
[17] to $q$-colored partitions.

The ESF appears in connection with a variety of models. In order to illustrate
the analogies between $E_{\theta,\mathbf{p}}^{n}$ and $E_{\theta}$, let us
briefly discuss two of them: Ewens’ original allelic partition, and the CRP.
In §4.1 we present in full detail the polychromatic analogue to Hoppe’s urn
model [13].

##### The ESF in population genetics

In the seminal work [9], W.J. Ewens introduced the formula later named after
him, and showed that $E_{\theta}$ is the joint probability distribution of the
number of selectively neutral alleles $A_{i}^{\scriptscriptstyle{(n)}}$
represented $i$ times in a sample of $n$ genes taken from a large $(\gg n)$
population, viz.

$\mathbf{P}[A^{\scriptscriptstyle{(n)}}_{1}=\lambda_{1},\dotsc,A^{\scriptscriptstyle{(n)}}_{n}=\lambda_{n}]=E_{\theta}(\boldsymbol{\lambda})\,\mathrm{,}\;\,$

where the parameter $\theta>0$ defines the rate $\tfrac{\theta}{\theta+n}$ at
which novel alleles appear.

The polychromatic analogue $E_{\theta,\mathbf{p}}^{n}$ to the ESF is the
distribution of the very same model, when alleles are additionally marked by a
‘color’ in $[q]$. Such a marking describes any of $q$ (hereditary or non-
hereditary) features specific to a given allele and which are not reflected by
the sequence of its base pairs. This includes, for instance, _in situ_
epigenetic modifications such as DNA-methylation.

##### Tourists at the Chinese restaurant

It would not be difficult to introduce polychromatic generalizations to many
well-known problems and constructions in the theory, such as the _Spaghetti
Loop distribution_ , or the _Feller coupling_. For the sake of brevity, we
only discuss the _Chinese restaurant process_ (CRP). In [1], D.J. Aldous
introduced333In fact, Aldous credits the introduction of the CRP to J. Pitman,
who in turn acknowledges the contribution of L. Dubins, see e.g. the
attribution to Dubins and Pitman in [28, §4.1]. the CRP as a sequential
description of the sampling of random partitions distributed according to the
Poisson–Dirichlet distribution. The process (and many of its variations) has
proven a very successful tool in the study of random partitions/permutations.
Let us briefly discuss a variation444A variation which may already be familiar
to those non-Chinese speaking customers ordering blindly from the “secret
menu”. of the CRP well-suited to describe our colored partitions.

As usual, « _[customers] $1,2,\dotsc,n$ arrive sequentially at an initially
empty restaurant with a large number of large [circular] tables. [Customer]
$j$ either sits at the same table as [customer] $i$, with probability
$1/(j-1+\theta)$ for each $i<j$, or else sits at an empty table, with
probability $\theta/(j-1+\theta)$_.» [1, (11.19), p. 91]. Additionally
however, each customer randomly chooses to order from one out of the $q$
proposed menus, independently of the other customers and according to a fixed
categorical distribution with parameter $\mathbf{p}$. The colored partition
‘people at each table ordering from each menu’ is distributed according to
$E_{\theta,\mathbf{p}}^{n}$.

##### Plan of the work

In §2.1 we introduce some necessary notation and define the pattern inventory
$\mathscr{Z}_{\mathbf{n}}$ in the right-hand side of (1.1). In §2.2 we show
that $\mathscr{Z}_{\mathbf{n}}$ coincides with a ‘refined’ cycle index
polynomial $Z_{\mathbf{n}}$ of a certain group action, counting $q$-colored
partitions coloring $\mathbf{n}$. We then move to prove Theorem Theorem 1
(§3.4) together with an overview of previously known results (§3.1), some
corollaries (§3.2), and applications to other measures (§3.3). Finally, we
study the polychromatic ESF by means of a polychromatic Hoppe urn model (§4.1)
and discuss its consistency in the sense of Kingman (§4.2).

##  2 Counting pattern inventories

For $n\in{\mathbb{N}}_{1}$ let
$[n]\operatorname{\coloneqq}\left\\{1,\dotsc,n\right\\}$, and
$\mathfrak{S}_{n}$ be the symmetric group of degree $n$, naturally acting on
$[n]$ by permutation of its elements.

##### Multisets

Given a set $S$, an _$S$ -multiset_ is any map $m\colon S\to{\mathbb{N}}_{0}$.
We denote by $\mathsf{supp}\,{m}$ the _support_ of $m$. The _cardinality_
${\mathsf{card}(m)}$ of $m$ is the sum $\sum_{s\in S}m(s)$ of all its values.
Given a map $f\colon S\to T$, the _push-forward via $f$_ of an $S$-multiset
$m$ is the $T$-multiset

$f_{*}m\operatorname{\coloneqq}\sum_{s\in\mathsf{supp}\,{m}}m(s)\operatorname{\mathbf{1}}_{f(s)}\,\mathrm{.}$
(2.1)

##### Vectors

Whenever no confusion may arise, we do not distinguish between row vectors and
column vectors. When relevant, we write $\mathbf{x}^{\scriptscriptstyle{(n)}}$
to indicate that $\mathbf{x}\in{\mathbb{R}}^{n}$ or, more generally, that
$\mathbf{x}$ has $n$ entries. Let $\mathbf{e}_{i}^{\scriptscriptstyle{(n)}}$
be the $i^{\textrm{th}}$ vector of the canonical basis of ${\mathbb{R}}^{n}$,
and set
${\mathbf{1}}^{\scriptscriptstyle{(n)}}\operatorname{\coloneqq}\left(1\right)_{i\in[n]}$
and analogously for $\operatorname{{\mathbf{0}}}^{{\scriptscriptstyle{(n)}}}$.
For vectors $\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}$ and
$\pi\in\mathfrak{S}_{n}$, write

$\displaystyle\mathbf{x}\cdot\mathbf{y}\operatorname{\coloneqq}$
$\displaystyle\ x_{1}y_{1}+\cdots+x_{n}y_{n}\,\mathrm{,}\;\,$
$\displaystyle\mathbf{x}\diamond\mathbf{y}\operatorname{\coloneqq}$
$\displaystyle\left(x_{1}y_{1},\dotsc,x_{n}y_{n}\right)\,\mathrm{,}\;\,$
$\displaystyle\mathbf{x}^{\diamond n}\operatorname{\coloneqq}$
$\displaystyle\left(x_{1}^{n},\dotsc,x_{n}^{n}\right)\,\mathrm{,}\;\,$
$\displaystyle{{}\mathbf{x}}_{\bullet}\operatorname{\coloneqq}$
$\displaystyle\ \mathbf{x}\cdot{\mathbf{1}}\,\mathrm{.}$

For any $f\colon{\mathbb{C}}\to{\mathbb{C}}$ further write
$f(\mathbf{x})\operatorname{\coloneqq}f(x_{1})\cdots f(x_{n})$.

##### Matrices

For a matrix
$\mathbf{M}\operatorname{\coloneqq}\big{[}m_{i,j}\big{]}_{i\in[a],j\in[b]}\in{\mathbb{R}}^{a\times
b}$ ($a$ rows, $b$ columns) set

$\displaystyle\mathbf{M}_{i}\operatorname{\coloneqq}$ $\displaystyle\
\mathbf{e}_{i}^{\scriptscriptstyle{(a)}}\mathbf{M}\,\mathrm{,}\;\,$
$\displaystyle\mathsf{row}(\mathbf{M})\operatorname{\coloneqq}$
$\displaystyle\
\mathbf{M}{\mathbf{1}}^{{\scriptscriptstyle{(b)}}}\in{\mathbb{R}}^{a}\,\mathrm{,}\;\,$
$\displaystyle\mathbf{M}^{j}\operatorname{\coloneqq}$ $\displaystyle\
\mathbf{M}\,\mathbf{e}_{j}^{{\scriptscriptstyle{(b)}}}\,\mathrm{,}\;\,$
$\displaystyle\mathsf{col}(\mathbf{M})\operatorname{\coloneqq}$
$\displaystyle\
{\mathbf{1}}^{\scriptscriptstyle{(a)}}\mathbf{M}\in{\mathbb{R}}^{b}\,\mathrm{.}$

In words: $\mathbf{M}_{i}$ is the $i^{\textrm{th}}$ row of $\mathbf{M}$ and
$\mathbf{M}^{j}$ is the $j^{\textrm{th}}$ column of $\mathbf{M}$, while
$\mathsf{row}(\mathbf{M})$ is the vector of the rows’ lengths of $\mathbf{M}$
and $\mathsf{col}(\mathbf{M})$ is the vector of the columns’ lengths of
$\mathbf{M}$. For matrices $\mathbf{S},\mathbf{M}\in{\mathbb{R}}^{a\times b}$
set
$\mathbf{S}^{\mathbf{M}}\operatorname{\coloneqq}\prod_{i,j}^{a,b}s_{i,j}^{m_{i,j}}$
and $\mathbf{M}!\operatorname{\coloneqq}\prod_{i,j}^{a,b}m_{i,j}!$.

###  2.1 Colored partitions, permutations, patterns

Throughout, let $q,r\in{\mathbb{N}}_{1}$ and let
$\mathbf{n}\operatorname{\coloneqq}\left(n_{1},\dotsc,n_{q}\right)\in{\mathbb{N}}_{0}^{q}$.

######  Definition 2.1.

A _$q$ -coloring_ of $[{{}\mathbf{n}}_{\bullet}]$ is any function assigning to
each element of $[{{}\mathbf{n}}_{\bullet}]$ a color in $[q]$. An
_$\mathbf{n}$ -coloring_ is a $q$-coloring $\mathfrak{c}$ of
$[{{}\mathbf{n}}_{\bullet}]$ with $\mathfrak{c}^{-1}(j)=n_{j}$ for each
$j\in[q]$.

If not otherwise stated, everywhere in the following we fix an
$\mathbf{n}$-coloring $\mathfrak{c}_{\mathbf{n}}$. Many assertions will depend
on $\mathfrak{c}_{\mathbf{n}}$, but they do so only via the choice of
$\mathbf{n}$.

####  2.1.1 Colored partitions

Set
${\mathbb{N}}_{*}^{q}\operatorname{\coloneqq}{\mathbb{N}}_{0}^{q}\setminus\left\\{\operatorname{{\mathbf{0}}}^{\scriptscriptstyle{(q)}}\right\\}$.
For an ${\mathbb{N}}^{q}_{*}$-multiset $A$, we define the quantity

$\mathsf{col}(A)\operatorname{\coloneqq}\sum_{\mathbf{a}\in\mathsf{supp}A}A(\mathbf{a})\,\mathbf{a}\quad\in\overline{\mathbb{N}}_{0}^{q}\,\mathrm{.}$

######  Definition 2.2 (Colored partitions).

A _$q$ -colored partition of $\mathbf{n}$_ is any
${\mathbb{N}}_{*}^{q}$-multiset $A$ satisfying $\mathsf{col}(A)=\mathbf{n}$,
in which case we write $A\vDash\mathbf{n}$. We denote by
$\mathcal{A}_{\mathbf{n}}$ the family of all $q$-colored partitions of
$\mathbf{n}$. The _shape_ of a $q$-colored partition $A$ is the integer
partition $\mathsf{shape}(A)\vdash{{}\mathbf{n}}_{\bullet}$ given by

$\mathsf{shape}(A)_{i}\operatorname{\coloneqq}\sum_{\mathbf{a}\in\mathsf{supp}A\,:\,{{}\mathbf{a}}_{\bullet}=i}A(\mathbf{a})\,\mathrm{,}\;\,\qquad
i\in[{{}\mathbf{n}}_{\bullet}]\,\mathrm{.}$

######  Definition 2.3.

The _multinomial coefficient_ of a $q$-colored partition $A$ of $\mathbf{n}$
is

$\displaystyle M_{\mathbf{n}}(A)\operatorname{\coloneqq}$ $\displaystyle\
\mathbf{n}!\prod_{\mathbf{a}\in\mathsf{supp}A}\frac{\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}}{{{}\mathbf{a}}_{\bullet}^{A(\mathbf{a})}A(\mathbf{a})!}\,\mathrm{.}$
(2.2)

######  Remark 2.4 ($q=1$).

When $q=1$, every $1$-colored partition is identical to its shape. Letting
$n\operatorname{\coloneqq}n_{1}$, hence $\mathbf{n}=(n)$, the multinomial
coefficient of $A\in\mathcal{A}_{\mathbf{n}}$ reduces to the _multinomial
coefficient of the second kind_ of the integer partition
$\boldsymbol{\lambda}\operatorname{\coloneqq}\mathsf{shape}(A)\vdash n$, viz.

$\displaystyle
M_{\mathbf{n}}(A)=M_{2}(\boldsymbol{\lambda})\operatorname{\coloneqq}n!\prod_{i\in\mathsf{supp}A}\frac{1}{i^{\lambda_{i}}\lambda_{i}!}\,\mathrm{,}\;\,\qquad\lambda_{i}\operatorname{\coloneqq}A(i)\,\mathrm{.}$
(2.3)

####  2.1.2 Pattern inventory

Let $G<\mathfrak{S}_{n}$ be a permutation group of degree $n$. The _cycle
index polynomial_ $Z^{G}$ of $G$ is

$Z^{G}(\mathbf{t})\operatorname{\coloneqq}\frac{1}{\left\lvert
G\right\rvert}\sum_{\pi\in
G}\mathbf{t}^{\boldsymbol{\lambda}(\pi)}\,\mathrm{,}\;\,\qquad\mathbf{t}=\left(t_{1},\dotsc,t_{n}\right)\,\mathrm{,}\;\,$

where $\boldsymbol{\lambda}(\pi)\vdash n$ accounts for the number of cycles in
$\pi$ of given length, i.e. $\lambda_{1}(\pi)$ is the number of fixed points
of $\pi$, $\lambda_{2}(\pi)$ the number of $2$-cycles in $\pi$, and so on. We
denote by $Z_{n}\operatorname{\coloneqq}Z^{\mathfrak{S}_{n}}$ the cycle index
polynomial of $\mathfrak{S}_{n}$. It is not difficult to show that (cf. (2.3))

$Z_{n}(\mathbf{t})=\frac{1}{n!}\sum_{\boldsymbol{\lambda}\vdash
n}M_{2}(\boldsymbol{\lambda})\,\mathbf{t}^{\boldsymbol{\lambda}}\,\mathrm{,}\;\,\qquad\mathbf{t}=\left(t_{1},\dotsc,t_{n}\right)\,\mathrm{.}$
(2.4)

##### Pattern inventory

We represent a permutation $\pi$ in its cycle notation, viz.

$\pi=(y_{1,1}y_{1,2}\cdots)(y_{2,1}y_{2,2}\cdots)\cdots(y_{r,1}y_{r,2}\cdots)\,\mathrm{.}$
(2.5)

Let
$\mathbf{S}\operatorname{\coloneqq}\left(\mathbf{s}_{1},\dotsc,\mathbf{s}_{q}\right)$
be a $k\times q$-matrix of dummy variables. We denote by
$\mathbf{S}^{1}=\mathbf{s}_{1},\dotsc,\mathbf{S}^{q}=\mathbf{s}_{q}$ the
columns of $\mathbf{S}$ and by $\mathbf{S}_{1},\dotsc,\mathbf{S}_{k}$ the rows
of $\mathbf{S}$. Further let $\boldsymbol{\alpha}\in{\mathbb{R}}^{k}$.

The following definition is inspired by Pólya Enumeration Theory.

######  Definition 2.5 (Pattern inventory).

The $\mathbf{n}$-_pattern_ of a permutation $\pi$ is

$w_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}](\pi)\operatorname{\coloneqq}\prod_{i}^{r}\big{(}{\mathbf{s}_{\mathfrak{c}_{\mathbf{n}}(y_{i,1})}\diamond\mathbf{s}_{\mathfrak{c}_{\mathbf{n}}(y_{i,2})}\diamond\cdots}\big{)}\cdot\boldsymbol{\alpha}\,\mathrm{.}$

The _pattern inventory of $\mathbf{n}$_ is the polynomial

$\mathscr{Z}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\operatorname{\coloneqq}\frac{1}{\mathbf{n}!}\sum_{\pi\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}}w_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}](\pi)\,\mathrm{.}$
(2.6)

Up to a different normalization, $\mathscr{Z}_{\mathbf{n}}$ is a refinement of
the cycle index polynomial of $\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$, in
the sense that each monomial in $\mathscr{Z}_{\mathbf{n}}$ depends not only on
the cycle structure of a permutation, but also on its coloring. In order to
simplify the expression of $\mathscr{Z}_{\mathbf{n}}$, let

$\displaystyle Z_{\mathbf{n}}(\mathbf{t})\operatorname{\coloneqq}$
$\displaystyle\
\frac{1}{\mathbf{n}!}\sum_{A\vDash\mathbf{n}}M_{\mathbf{n}}(A)\,\prod_{\mathbf{a}\in\mathsf{supp}A}t_{\mathbf{a}}^{A(\mathbf{a})}\,\mathrm{,}\;\,\qquad\mathbf{t}\operatorname{\coloneqq}\left(t_{\mathbf{a}}\right)_{\mathbf{a}\leq_{\diamond}\mathbf{n}}\,\mathrm{.}$
(2.7)

Finally, for every $\mathbf{a}\leq_{\diamond}\mathbf{n}$ set

$\displaystyle\omega_{\mathbf{a}}[\mathbf{S};\boldsymbol{\alpha}]\operatorname{\coloneqq}\big{(}{\mathbf{s}_{1}^{\diamond
a_{1}}\diamond\cdots\diamond\mathbf{s}_{q}^{\diamond
a_{q}}}\big{)}\cdot\boldsymbol{\alpha}\,\mathrm{,}\;\,\quad\text{and}\quad\Omega_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\operatorname{\coloneqq}$
$\displaystyle\
\left(\omega_{\mathbf{a}}[\mathbf{S};\boldsymbol{\alpha}]\right)_{\mathbf{a}\leq_{\diamond}\mathbf{n}}\,\mathrm{.}$
(2.8)

In Theorem 2.14 below, we will prove that

$\displaystyle\mathscr{Z}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=Z_{\mathbf{n}}(\Omega_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}])\,\mathrm{.}$
(2.9)

######  Remark 2.6 ($q=1$).

When $q=1$, the polynomial $Z_{\mathbf{n}}$ in (2.7) reduces to $Z_{n}$ in
(2.4).

###  2.2 Group actions

In order to prove (2.9), we identify the algebraic meaning of
$\mathscr{Z}_{\mathbf{n}}$ in terms of the action of a certain group of
permutations.

####  2.2.1 Some bijections of the symmetric group

Let $G$ be any finite group. For $h\in G$ we denote by $\tau_{h}\colon G\to G$
the conjugation map $\tau_{h}\colon g\mapsto hgh^{-1}$. For each $\pi$ in
$\mathfrak{S}_{n}$ and $i,j\in[n]$ we write

$i\mathrel{\underset{\pi}{{[}1]{\sim}}}j\quad\text{if}\quad
j=\pi^{p}(i)\quad\text{for some
}p\in\operatorname{{\mathbb{Z}}}\,\mathrm{,}\;\,$

i.e., if $i,j\in[n]$ belong to the same orbit (cycle) of $\pi$. We note that
$\mathrel{\underset{\pi}{{[}1]{$\sim$}}}$ is an equivalence relation on $[n]$,
and that

$i\mathrel{\underset{\pi}{{[}1]{\sim}}}j\iff\sigma(i)\mathrel{\underset{\tau_{\sigma}(\pi)}{{[}1]{\sim}}}\sigma(j)\,\mathrm{,}\;\,\qquad
i,j\in[n]\,\mathrm{,}\;\,\quad\pi,\sigma\in\mathfrak{S}_{n}\,\mathrm{.}$
(2.10)

Let $(B_{n},\circ)$ be the group of bijections of $\mathfrak{S}_{n}$ leaving
conjugacy classes invariant. That is, $g\in B_{n}$ if and only if $g(\pi)$ has
the same cycle structure as $\pi$ for every $\pi\in\mathfrak{S}_{n}$. We have
$B_{n}\cong\bigtimes_{\boldsymbol{\lambda}\vdash
n}\mathfrak{S}_{M_{2}(\boldsymbol{\lambda})}$ in a natural way.

We denote by $H_{n}$ the subset of all $h\in B_{n}$ such that

$\displaystyle i\mathrel{\underset{\pi}{{[}1]{$\sim$}}}j\implies
i\mathrel{\underset{h(\pi)}{{[}1]{$\sim$}}}j\,\mathrm{,}\;\,\qquad
i,j\in[n]\,\mathrm{,}\;\,\quad\pi\in\mathfrak{S}_{n}\,\mathrm{.}$ (2.11)

Let $h_{1},h_{2}\in H_{n}$. Consecutive applications of (2.11) show that

$i\mathrel{\underset{\pi}{{[}1]{\sim}}}j\implies
i\mathrel{\underset{h_{1}(\pi)}{{[}1]{\sim}}}j\implies
i\mathrel{\underset{(h_{2}\circ
h_{1})(\pi)}{{[}1]{\sim}}}j\,\mathrm{,}\;\,\qquad
i,j\in[n]\,\mathrm{,}\;\,\pi\in\mathfrak{S}_{n}\,\mathrm{.}$

Thus, $H_{n}$ is closed under $\circ$ and therefore it is a subgroup of (the
_finite_ group) $B_{n}$.

In order to exemplify some elements of $H_{n}$, for
$\pi,\sigma\in\mathfrak{S}_{n}$, set

$\displaystyle
f_{\sigma,\pi}(i)\operatorname{\coloneqq}\sigma^{p_{\sigma,\pi}(i)}(i)\,\mathrm{,}\;\,\quad
p_{\sigma,\pi}(i)\operatorname{\coloneqq}\min\left\\{p\geq
1:\sigma^{p}(i)\mathrel{\underset{\pi}{{[}1]{$\sim$}}}i\right\\}\,\mathrm{,}\;\,\quad
i\in[n]\,\mathrm{.}$ (2.12)

For each $\pi\in\mathfrak{S}_{n}$ it is readily verified that
$f_{\sigma,\pi}\in\mathfrak{S}_{n}$ is a bijection and
$f_{\sigma,\pi}^{-1}=f_{\sigma^{-1},\pi}$.

######  Example 2.7.

For $\sigma\in\mathfrak{S}_{n}$ define $h_{\sigma}$ by
$h_{\sigma}\colon\pi\mapsto\tau_{f_{\sigma,\pi}}(\pi)$. Then, $h_{\sigma}\in
H_{n}$ and $h_{\sigma}^{-1}=h_{\sigma^{-1}}$.

###### Proof.

We verify that $h_{\sigma}\in H_{n}$. Firstly, since
$f_{\sigma,\pi}^{-1}=f_{\sigma^{-1},\pi}$, we have that $h_{\sigma}$ is a
bijection with inverse $h_{\sigma^{-1}}$. By definition, $h_{\sigma}(\pi)$ is
conjugate to $\pi$ for every $\pi\in\mathfrak{S}_{n}$. Thus, $h_{\sigma}\in
B_{n}$ and it suffices to show (2.11). Furthermore,
$f_{\sigma,\pi}(i)\mathrel{\underset{\pi}{{[}1]{$\sim$}}}i$ for every $i$ by
the definition (2.12) of $f_{\sigma,\pi}$. Thus, $\pi$ and $h_{\sigma}(\pi)$
define the same equivalence relation, viz.

$i\mathrel{\underset{\pi}{{[}1]{\sim}}}j\iff
i\mathrel{\underset{h_{\sigma}(\pi)}{{[}1]{\sim}}}j\,\mathrm{,}\;\,\quad
i,j\in[n]\,\mathrm{,}\;\,\qquad\pi\in\mathfrak{S}_{n}\,\mathrm{,}\;\,$ (2.13)

which implies (2.11). ∎

######  Remark 2.8.

For $n\geq 4$, the map $\sigma\mapsto h_{\sigma}$ is _not_ a group
homomorphism. For instance, choose $\sigma_{1}\operatorname{\coloneqq}(24)$,
$\sigma_{2}\operatorname{\coloneqq}(34)$, and $\pi=(123)$, and notice that
$(h_{\sigma_{1}}\circ
h_{\sigma_{2}})(\pi)=\pi\neq(132)=h_{\sigma_{1}\sigma_{2}}(\pi)$.

######  Remark 2.9.

$(a)$ For every $n$, every $h\in H_{n}$ fixes transpositions, hence
$H_{1},H_{2}$ are the trivial group, and $H_{3}=\mathfrak{S}_{2}$ is the group
with non-trivial element exchanging the $3$-cycles in $\mathfrak{S}_{3}$.
$(b)$ For $n\geq 3$, the group $H_{n}$ is not a subgroup of the automorphism
group of $\mathfrak{S}_{n}$. Indeed, elements of $h$ are in general _not_
group homomorphisms. For example, let $h\in H_{n}$ be the map exchanging
$(123)$ and $(132)$ and fixing all other permutations. Assuming $h$ is a
homomorphism we would have
$(12)=(123)(23)=h\big{(}{(132)}\big{)}h\big{(}{(23)}\big{)}=h\big{(}{(132)(23)}\big{)}=h\big{(}{(13)}\big{)}\,\mathrm{,}\;\,$

which contradicts a.  $(c)$ $H_{n}$is not normal in $B_{n}$ for $n\geq 4$.
Indeed, let $f\in B_{n}$ be the map exchanging $(123)$ with $(124)$ and fixing
all other permutations, and $h\operatorname{\coloneqq}h_{(123)}$. Then,

$\big{(}{\tau_{f}(h)}\big{)}\big{(}{(123)}\big{)}=(214)\,\mathrm{,}\;\,$

which violates (2.11) with $\tau_{f}(h)$ in place of $h$.

Whereas $H_{n}$ is not a normal subgroup of $B_{n}$, we still have the
following.

######  Lemma 2.10.

Let $\sigma\in\mathfrak{S}_{n}$. Then,

$\varphi_{\sigma}\operatorname{\coloneqq}\tau_{\tau_{\sigma}}\colon
h\longmapsto\tau_{\sigma}\circ h\circ\tau_{\sigma}^{-1}$ (2.14)

is an automorphism of $H_{n}$, and
$\varphi_{\,\cdot\,}\colon\mathfrak{S}_{n}\to\mathrm{Aut}(H_{n})$ is a group
homomorphism.

###### Proof.

Since $\tau_{\sigma}$ leaves conjugacy classes in $\mathfrak{S}_{n}$
invariant, we have $\tau_{\sigma}\in B_{n}$. Thus $\varphi_{\sigma}$ is an
inner automorphism of $B_{n}$. Furthermore, since for every group $G$ the map
$\tau^{G}\colon g\mapsto\tau_{g}$ is a group homomorphism
$G\to\mathrm{Aut}(G)$, the map
$\varphi_{\,\cdot\,}=\tau^{B_{n}}\circ\tau^{\mathfrak{S}_{n}}$ is a group
homomorphism as well. Thus, it suffices to show that
$\varphi_{\sigma}(H_{n})\subset H_{n}$ for every $\sigma\in\mathfrak{S}_{n}$.
To this end, it suffices to verify (2.11) with $\varphi_{\sigma}(h)$ in place
of $h$. Indeed, respectively by (2.10) with $\sigma^{-1}$ in place of
$\sigma$, by (2.11), and by (2.10),

$\displaystyle i\mathrel{\underset{\pi}{{[}1]{$\sim$}}}j\implies$
$\displaystyle\
\sigma^{-1}(i)\mathrel{\underset{\tau_{\sigma^{-1}}(\pi)}{{[}1]{$\sim$}}}\sigma^{-1}(j)\implies\sigma^{-1}(i)\mathrel{\underset{(h\circ\tau_{\sigma^{-1}})(\pi)}{{[}1]{$\sim$}}}\sigma^{-1}(j)$
$\displaystyle\implies$ $\displaystyle\
\sigma\sigma^{-1}(i)\mathrel{\underset{(\tau_{\sigma}\circ
h\circ\tau_{\sigma^{-1}})(\pi)}{{[}1]{$\sim$}}}\sigma\sigma^{-1}(j)$

and the conclusion follows since $\tau_{\sigma}^{-1}=\tau_{\sigma^{-1}}$. ∎

####  2.2.2 Semi-direct product and group action

Fix an $\mathbf{n}$-coloring $\mathfrak{c}_{\mathbf{n}}$. All results in the
following hold for every such coloring. Proposition 2.13 below will provide an
algebraic interpretation of the multinomial coefficient $M_{\mathbf{n}}$ in
(2.2) by means of the surjective map
$\Pi=\Pi_{\mathfrak{c}_{\mathbf{n}}}\colon\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}\to\mathcal{A}_{\mathbf{n}}$
which we now define. Firstly, to every _cycle_ $\kappa=(y_{1}y_{2}\cdots)$ we
associate a vector $\boldsymbol{\varepsilon}(\kappa)$ in
${\mathbb{N}}_{0}^{q}$ by

$\boldsymbol{\varepsilon}(\kappa)_{j}\operatorname{\coloneqq}\left\lvert\left\\{h:\mathfrak{c}_{\mathbf{n}}(y_{h})=j\right\\}\right\rvert\,\mathrm{,}\;\,\qquad
j\in[q]\,\mathrm{.}$ (2.15)

For
$\pi=\kappa_{1}\cdots\kappa_{r}\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$,
with cycles $\kappa_{1},\dotsc,\kappa_{r}$ (including fixed points), we then
set

$\Pi\colon\pi\longmapsto\sum_{i=1}^{r}\operatorname{\mathbf{1}}_{\boldsymbol{\varepsilon}(\kappa_{i})}\,\mathrm{.}$
(2.16)

##### Semi-direct product

In the following, we regard

$\mathfrak{S}_{\mathbf{n}}\operatorname{\coloneqq}\mathfrak{S}_{\mathfrak{c}_{\mathbf{n}}^{-1}(1)}\times\cdots\times\mathfrak{S}_{\mathfrak{c}_{\mathbf{n}}^{-1}(q)}\cong\mathfrak{S}_{n_{1}}\times\cdots\times\mathfrak{S}_{n_{q}}$

as a subgroup of $\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$.

######  Definition 2.11.

Let
$(G_{\mathbf{n}},\star)\operatorname{\coloneqq}H_{{{}\mathbf{n}}_{\bullet}}\rtimes\mathfrak{S}_{\mathbf{n}}$
be the semi-direct product induced by the group homomorphism
$\varphi_{\,\cdot\,}$ defined by (2.14), that is

$(h_{1},\sigma_{1})\star(h_{2},\sigma_{2})\operatorname{\coloneqq}\
(h_{1}\circ\varphi_{\sigma_{1}}(h_{2}),\sigma_{1}\sigma_{2})\,\mathrm{.}$

######  Lemma 2.12.

The function $\,\raisebox{0.0pt}{{[}-1]{$\circlearrowleft$}}\,\colon
G_{\mathbf{n}}\times\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}\to\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$
given by

$\displaystyle\,\raisebox{0.0pt}{{[}-1]{$\circlearrowleft$}}\,\colon\big{(}{(h,\sigma),\pi}\big{)}\longmapsto(h,\sigma).\pi\operatorname{\coloneqq}\,(h\circ\tau_{\sigma})(\pi)$
(2.17)

defines a group action of $G_{\mathbf{n}}$ on
$\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$, faithful if
${{}\mathbf{n}}_{\bullet}\geq 3$.

###### Proof.

In order to show that [-1]$\circlearrowleft$ is a group action it suffices to
verify that

$\displaystyle\big{(}{(h_{1},\sigma_{1})\star(h_{2},\sigma_{2})}\big{)}.\pi=$
$\displaystyle\
\big{(}{h_{1}\circ\varphi_{\sigma_{1}}(h_{2})}\big{)}(\sigma_{1}\sigma_{2}\pi\sigma_{2}^{-1}\sigma_{1}^{-1})$
$\displaystyle=$ $\displaystyle\
h_{1}\big{(}{\sigma_{1}h_{2}(\sigma_{2}\pi\sigma_{2}^{-1})\sigma_{1}^{-1}}\big{)}$
$\displaystyle=$ $\displaystyle\
(h_{1},\sigma_{1}).(h_{2},\sigma_{2}).\pi\,\mathrm{.}$

In order to show faithfulness, it suffices to prove that
$(h,\sigma)=(\mathrm{id},e)$ whenever

$(h,\sigma).\pi=\pi\,\mathrm{,}\;\,\qquad\pi\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}\,\mathrm{.}$
(2.18)

If $\sigma=e$, since $B_{{{}\mathbf{n}}_{\bullet}}$ (hence
$H_{{{}\mathbf{n}}_{\bullet}}$) acts faithfully on
$\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$, (2.18) implies $h=\mathrm{id}$. If
$\sigma\neq e$, since ${{}\mathbf{n}}_{\bullet}\geq 3$, there exist mutually
different $i,j,k\in[n]$ with $\sigma(i)=j$. Choosing
$\pi\operatorname{\coloneqq}(ik)$,

$(h,\sigma).\pi=h\big{(}{(\sigma(i),\sigma(k))}\big{)}=h\big{(}{(j,\sigma(k))}\big{)}=(j,\sigma(k))\neq\pi\,\mathrm{,}\;\,$

where the last equality follows again from Remark 2.9. ∎

######  Proposition 2.13.

The orbit space $\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}/G$ is (parametrized
by) the set $\mathcal{A}_{\mathbf{n}}$ of all $q$-colored partitions, and
$\left\lvert G.\pi\right\rvert=M_{\mathbf{n}}(\Pi(\pi))$ for every
$\pi\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$.

###### Proof.

For every $\pi,\pi^{\prime}\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$, let us
prove that $\Pi(\pi)=\Pi(\pi^{\prime})$ if and only if $\pi\in
G.\pi^{\prime}$. Let $\pi=\kappa_{1}\cdots\kappa_{r}$ and
$\pi^{\prime}=\kappa_{1}^{\prime}\cdots\kappa_{r^{\prime}}^{\prime}$ be cycle
decompositions. If $\Pi(\pi)=\Pi(\pi^{\prime})$, then $r=r^{\prime}$ and, up
to reordering the cycles, we may assume without loss of generality that
$\Pi(\kappa_{i})=\Pi(\kappa_{i}^{\prime})$ for every $i$. Therefore, there
exists $\sigma\in\mathfrak{S}_{\mathbf{n}}$ such that for every $i$ the cycles
$\kappa_{i}$ and $\sigma\kappa_{i}^{\prime}\sigma^{-1}$ transitively permute
the same set of numbers. Equivalently,

$i\mathrel{\underset{\pi}{{[}1]{\sim}}}j\iff
i\mathrel{\underset{\tau_{\sigma}(\pi^{\prime})}{{[}1]{\sim}}}j\,\mathrm{,}\;\,\qquad
i,j\in[{{}\mathbf{n}}_{\bullet}]\,\mathrm{.}$

Hence, the map $h\in B_{n}$ that swaps $\pi$ and
$\tau_{\sigma}(\pi^{\prime})$, and fixes every other element of
$\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$ is in
$H_{{{}\mathbf{n}}_{\bullet}}$. We can thus write
$\pi=(h,\sigma).\pi^{\prime}$. Conversely, if $\pi=(h,\sigma).\pi^{\prime}$
holds for some $h$ and $\sigma$, then we can rearrange the cycle
decompositions $\pi=\kappa_{1}\cdots\kappa_{r}$ and
$\tau_{\sigma}(\pi^{\prime})=\tau_{\sigma}(\kappa^{\prime}_{1})\cdots\tau_{\sigma}(\kappa^{\prime}_{r})$
in such a way that $\kappa_{i}$ and $\sigma\kappa_{i}^{\prime}\sigma^{-1}$
transitively permute the same set of numbers for every $i$. Therefore,
$\Pi(\kappa_{i})=\Pi(\sigma\kappa_{i}^{\prime}\sigma^{-1})$. Furthermore,
since $\sigma\in\mathfrak{S}_{\mathbf{n}}$, we have
$\Pi(\sigma\kappa_{i}^{\prime}\sigma^{-1})=\Pi(\kappa_{i}^{\prime})$, whence
$\Pi(\kappa_{i})=\Pi(\kappa_{i}^{\prime})$ as desired.

##### Cardinality of the orbits

Let $A\in\mathcal{A}_{\mathbf{n}}$. We aim to show that
$\left\lvert\Pi^{-1}(A)\right\rvert=M_{\mathbf{n}}(A)$. In order to do so, it
is convenient to introduce some new sets and maps, as schematized in Figure 1
below, and compute the cardinality of their fibers.

  1. $(a)$

Firstly, given a vector
$\mathbf{c}\operatorname{\coloneqq}\left(c_{1},c_{2},\dotsc\right)$ with
entries in $[q]$ and arbitrary (possibly zero) length, we consider the
${\mathbb{N}}_{0}^{q}$-valued map $\boldsymbol{\varepsilon}$ defined by

$\boldsymbol{\varepsilon}(\mathbf{c})_{j}\operatorname{\coloneqq}\left\lvert\left\\{h:c_{h}=j\right\\}\right\rvert\,\mathrm{,}\;\,\qquad
j\in[q]\,\mathrm{.}$ (2.19)

  2. $(b)$

We denote by $\\#\mathbf{M}$ the number of rows of a matrix $\mathbf{M}$. The
map $\\#$ is naturally extended to matrix-valued functions by post-
composition.

  3. $(c)$

Let $\mathcal{Y}$ be the space of all matrix-valued functions $Y$ on
${\mathbb{N}}^{q}_{*}$ satisfying, for all
$\mathbf{a}\in{\mathbb{N}}^{q}_{*}$,

$Y(\mathbf{a})_{i}\in\boldsymbol{\varepsilon}^{-1}(\mathbf{a})\,\mathrm{,}\;\,\quad
i\in[\\#Y(\mathbf{a})]\,\mathrm{,}\;\,\qquad\text{and}\qquad\\#\circ
Y\in\mathcal{A}_{\mathbf{n}}\,\mathrm{.}$

We explicitly allow for $Y(\mathbf{a})$ to possibly be the empty matrix for
some $\mathbf{a}\in{\mathbb{N}}^{q}_{*}$.

  4. $(d)$

Denote by $\mathfrak{c}_{\mathbf{n}}^{\diamond}$ the entry-by-entry extension
of $\mathfrak{c}_{\mathbf{n}}$ to vectors and matrices. We define
$\mathcal{X}$ as the set of all matrix-valued functions $X$ on
${\mathbb{N}}_{*}^{q}$,

$X(\mathbf{a})=\begin{bmatrix}y_{\mathbf{a},1,1}&y_{\mathbf{a},1,2}&\dotsc\\\
y_{\mathbf{a},2,1}&y_{\mathbf{a},2,2}&\dotsc\\\
\vdots&\vdots&\ddots\end{bmatrix}\,\mathrm{,}\;\,$

satisfying, for all $\mathbf{a}\in{\mathbb{N}}^{q}_{*}$,

$\displaystyle
X(\mathbf{a})_{i}\in(\boldsymbol{\varepsilon}\circ\mathfrak{c}_{\mathbf{n}}^{\diamond})^{-1}(\mathbf{a})\,\mathrm{,}\;\,\quad
i\in[\\#X(\mathbf{a})]\,\mathrm{,}\;\,$
$\displaystyle\left\\{y_{\mathbf{a},i,j}\right\\}_{\mathbf{a},i,j}=[{{}\mathbf{n}}_{\bullet}]\,\mathrm{,}\;\,\qquad\text{and}\qquad
y_{\mathbf{a},i,j}\neq
y_{\mathbf{a}^{\prime},i^{\prime},j^{\prime}}\,\mathrm{,}\;\,\quad\left(\mathbf{a},i,j\right)\neq\left(\mathbf{a}^{\prime},i^{\prime},j^{\prime}\right)\,\mathrm{.}$

  5. $(e)$

Denote by $\mathcal{Z}$ the family of set-valued functions of the form

$Z\colon\mathbf{a}\longmapsto\big{\\{}\left(y_{\mathbf{a},1,1},y_{\mathbf{a},1,2},\dotsc\right),\left(y_{\mathbf{a},2,1},y_{\mathbf{a},2,2},\dotsc\right),\dotsc\big{\\}}$
(2.20)

additionally so that

$\left(\mathbf{a}\longmapsto\begin{bmatrix}y_{\mathbf{a},1,1}&y_{\mathbf{a},1,2}&\dotsc\\\
y_{\mathbf{a},2,1}&y_{\mathbf{a},2,2}&\dotsc\\\
\vdots&\vdots&\ddots\end{bmatrix}\right)\in\mathcal{X}\,\mathrm{.}$

  6. $(f)$

Finally let $f_{1}\colon\mathcal{X}\to\mathcal{Z}$ and
$f_{2}\colon\mathcal{Z}\to\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$ be maps
_forgetting_ part of the structure:

$f_{1}(X)(\mathbf{a})\operatorname{\coloneqq}\left\\{X(\mathbf{a})_{i}\right\\}_{i\in[\\#X(\mathbf{a})]}\,\mathrm{,}\;\,\qquad\mathbf{a}\in{\mathbb{N}}_{*}^{q}\,\mathrm{,}\;\,$

and, using the notation of (2.20),

$f_{2}\colon
Z\longmapsto\pi\operatorname{\coloneqq}\prod_{\begin{subarray}{c}\mathbf{a}\in{\mathbb{N}}_{*}^{q}\\\
Z(\mathbf{a})\neq\operatorname{\varnothing}\end{subarray}}\left(y_{\mathbf{a},1,1}\
y_{\mathbf{a},1,2}\ \cdots\right)\left(y_{\mathbf{a},2,1}\ y_{\mathbf{a},2,2}\
\cdots\right)\cdots\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}\,\mathrm{.}$

It is a tedious verification that the diagram in Figure 1 commutes.

${\mathcal{X}}$${\mathcal{Z}}$${\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}}$${\mathcal{Y}}$${\mathcal{A}_{\mathbf{n}}}$$\scriptstyle{f_{1}}$$\scriptstyle{\mathfrak{c}_{\mathbf{n}}^{\diamond}}$$\scriptstyle{f_{2}}$$\scriptstyle{\Pi}$$\scriptstyle{\\#}$
Figure 1: Auxiliary maps and sets in the proof of Proposition 2.13.

Now, let
$\pi=(y_{1,1}y_{1,2}\cdots)\cdots(y_{r,1}y_{r,2}\cdots)\in\Pi^{-1}(A)$ and
define $\mathbf{a}_{i}\leq_{\diamond}\mathbf{n}$ by

$\Pi\big{(}{(y_{i,1}y_{i,2}\cdots)}\big{)}=\operatorname{\mathbf{1}}_{\mathbf{a}_{i}}\,\mathrm{,}\;\,\qquad
i\in[r]\,\mathrm{.}$

The fiber $f_{2}^{-1}(\pi)$ consists of all the (distinct) set-valued
functions

$Z_{k_{1},\dotsc,k_{r}}\colon\mathbf{a}\longmapsto\big{\\{}{\big{(}\pi^{k_{i}}(y_{i,1}),\pi^{k_{i}}(y_{i,2}),\dotsc\big{)}}\big{\\}}_{i\colon\mathbf{a}=\mathbf{a}_{i}}\,\mathrm{,}\;\,\qquad
k_{1}\in[{{}\mathbf{a}_{1}}_{\bullet}],\dotsc,k_{r}\in[{{}\mathbf{a}_{r}}_{\bullet}]\,\mathrm{,}\;\,$

and has therefore cardinality $\left\lvert
f_{2}^{-1}(\pi)\right\rvert={{}\mathbf{a}_{1}}_{\bullet}\cdots{{}\mathbf{a}_{r}}_{\bullet}=\prod_{\mathbf{a}\in\mathsf{supp}A}{{}\mathbf{a}}_{\bullet}^{A(\mathbf{a})}$.
As for the fibers of $f_{1}$, given $Z\in(\Pi\circ f_{2})^{-1}(A)$ and $X\in
f_{1}^{-1}(Z)$, every element of $f_{1}^{-1}(Z)$ is induced by a permutation-
valued function $\varsigma$ on ${\mathbb{N}}_{*}^{q}$ such that

$\varsigma\colon\mathbf{a}\longmapsto\varsigma_{\mathbf{a}}\in\mathfrak{S}_{A(\mathbf{a})}\,\mathrm{,}\;\,\qquad\mathbf{a}\in{\mathbb{N}}_{*}^{q}\,\mathrm{,}\;\,$

via the formula

$X_{\varsigma}\colon\mathbf{a}\longmapsto
P_{\varsigma_{\mathbf{a}}}X(\mathbf{a})\,\mathrm{.}$

where $P_{\varsigma_{\mathbf{a}}}$ is the permutation matrix induced by
$\varsigma_{\mathbf{a}}$. It follows that $\left\lvert
f_{1}^{-1}(Z)\right\rvert=\prod_{\mathbf{a}\in\mathsf{supp}A}A(\mathbf{a})!$.
It is easy to see that the fibers of
$\mathfrak{c}_{\mathbf{n}}^{\diamond}\colon\mathcal{X}\to\mathcal{Y}$ all have
cardinality $\mathbf{n}!$. Lastly, the computation of the cardinality of the
fibers of $\\#\colon\mathcal{Y}\to\mathcal{A}_{\mathbf{n}}$ can be performed
$\mathbf{a}$ by $\mathbf{a}$ and, thanks to the properties of the multinomial
coefficient,

$\left\lvert\\#^{-1}(A)\right\rvert=\prod_{\mathbf{a}\in\mathsf{supp}A}\left\lvert\boldsymbol{\varepsilon}^{-1}(\mathbf{a})\right\rvert^{A(\mathbf{a})}=\prod_{\mathbf{a}\in\mathsf{supp}A}\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}\,\mathrm{.}$

In conclusion,

$\displaystyle\mathbf{n}!\prod_{\mathbf{a}\in\mathsf{supp}A}\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}=$
$\displaystyle\
\left\lvert(\\#\circ\mathfrak{c}_{\mathbf{n}}^{\diamond})^{-1}(A)\right\rvert=\sum_{\pi\in\Pi^{-1}(A)}\left\lvert(f_{2}\circ
f_{1})^{-1}(\pi)\right\rvert$ $\displaystyle=$ $\displaystyle\
\left\lvert\Pi^{-1}(A)\right\rvert\prod_{\mathbf{a}\in\mathsf{supp}A}{{}\mathbf{a}}_{\bullet}^{A(\mathbf{a})}A(\mathbf{a})!\,\mathrm{,}\;\,$

which yields the desired identity. ∎

We conclude this section with the proof of (2.9).

######  Theorem 2.14.

The polynomial $Z_{\mathbf{n}}$ in (2.7) is the orbit generating function of
the action (2.17). Furthermore, (2.9) holds.

###### Proof.

It suffices to collect all terms in $\mathscr{Z}_{\mathbf{n}}$ with the same
monomials. By Proposition 2.13, for each
$\pi\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$ there are exactly $\left\lvert
G.\pi\right\rvert=M_{\mathbf{n}}(\Pi(\pi))$ monomials indexed by $A=\Pi(\pi)$,
and the conclusion follows using that
$w_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}](\pi)=\prod_{\mathbf{a}\in\mathsf{supp}A}\omega_{\mathbf{a}}[\mathbf{S};\boldsymbol{\alpha}]^{A(\mathbf{a})}$.
∎

####  2.2.3 Necklaces

Theorem 2.14 provides an algebraic interpretation for (2.9). Let us now give a
combinatorial interpretation of the same formula, i.e. of the multinomial
coefficient $M_{\mathbf{n}}$, in terms of necklaces, which will in turn
provide a connection to ESF via the extension of the CRP discussed in §1.

On the one hand, waiters in our busy restaurant take care to remember, for
every table, which clients order from each menu. The arrangement of the
customers around the table is important in serving them efficiently. All the
information the waiters need about the customers’ arrangement is thus
contained in a $q$-colored necklace. On the other hand, chefs in the
restaurant only care about how many customers at each table order from each
menu, so that customers at the same table may be served at the same time. All
the information the chefs need about the customers’ arrangement is thus
contained in a $q$-colored partition. Let us now count $q$-colored partitions
by collecting together $q$-colored necklaces with the same occurrences of each
color.

For integer $q\in{\mathbb{N}}_{1}$ denote by $[q]^{*}$ the free monoid
generated by $[q]$. Elements of $[q]^{*}$ are called ($q$-)_words_. Two words
$u,v$ are _conjugate_ if there exist words $s,t$ so that $u=st$ and $v=ts$.
Two conjugate words are cyclic shifts of one another. Thus, conjugacy is an
equivalence relation on words. Its equivalence classes are called
($q$-)_necklaces_.

Let $\nu=\llbracket w\rrbracket$ be a necklace and $w=c_{1}c_{2}\cdots
c_{\ell}$ be any of its representatives. The _length_ $\ell_{\nu}$ of $\nu$ is
the total number $\ell$ of characters in $w$. The _period_ $p_{\nu}$ of $\nu$
is the minimal integer $p\geq 1$ with $c_{i}=c_{i+p-1\pmod{\ell}+1}$ for every
$i\in[\ell]$. Clearly, $p_{\nu}$ divides $\ell_{\nu}$.

  1. $(a)$

Let $w=c_{1}c_{2}\cdots\in[q]^{*}$. Consistently with (2.19), we denote by
$\boldsymbol{\varepsilon}(w)\in{\mathbb{N}}_{0}^{q}$ the vector of occurrences
of its characters, viz.

$\boldsymbol{\varepsilon}(w)_{j}\operatorname{\coloneqq}\left\lvert\left\\{h:c_{h}=j\right\\}\right\rvert\,\mathrm{.}$

It is readily seen that $\boldsymbol{\varepsilon}$ descends to a (non-
relabeled) map on necklaces.

  2. $(b)$

Let $\mathcal{N}_{\mathbf{n}}$ be the family of all multisets $N$ of
$q$-necklaces satisfying
$\boldsymbol{\varepsilon}_{*}N\in\mathcal{A}_{\mathbf{n}}$, cf. (2.1).

  3. $(c)$

Define a map $\mathfrak{c}_{\mathbf{n}}^{\diamond}$ on
$\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$ in the following way. For a cyclic
permutation $\kappa=\left(y_{1}y_{2}\cdots\right)$ let $\nu$ be the necklace
$\llbracket\mathfrak{c}_{\mathbf{n}}(y_{1})\ \mathfrak{c}_{\mathbf{n}}(y_{2})\
\cdots\rrbracket$ and set
$\mathfrak{c}_{\mathbf{n}}^{\diamond}(\kappa)\operatorname{\coloneqq}\operatorname{\mathbf{1}}_{\nu}$.
Extend $\mathfrak{c}_{\mathbf{n}}^{\diamond}$ by

$\mathfrak{c}_{\mathbf{n}}^{\diamond}\colon\pi\operatorname{\coloneqq}\kappa_{1}\cdots\kappa_{r}\longmapsto\sum_{i=1}^{r}\mathfrak{c}_{\mathbf{n}}^{\diamond}(\kappa_{i})\,\mathrm{.}$

  4. $(d)$

It is readily verified that
$\Pi=\boldsymbol{\varepsilon}_{*}\circ\mathfrak{c}_{\mathbf{n}}^{\diamond}\colon\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}\to\mathcal{A}_{\mathbf{n}}$
factors over $\mathcal{N}_{\mathbf{n}}$.

######  Proposition 2.15.

It holds that

$\left\lvert(\mathfrak{c}_{\mathbf{n}}^{\diamond})^{-1}(N)\right\rvert=\mathbf{n}!\prod_{\nu\in\mathsf{supp}N}\frac{p_{\nu}/\ell_{\nu}}{N(\nu)!}\qquad\text{and}\qquad
M_{\mathbf{n}}(A)=\mathbf{n}!\sum_{\begin{subarray}{c}N\in\mathcal{N}_{\mathbf{n}}\\\
\boldsymbol{\varepsilon}_{*}N=A\end{subarray}}\prod_{\nu\in\mathsf{supp}N}\frac{p_{\nu}/\ell_{\nu}}{N(\nu)!}\,\mathrm{.}$

###### Proof.

We provide a sketch of the proof, the details being similar to Proposition
2.13.

  1. $(a)$

A word in $[{{}\mathbf{n}}_{\bullet}]^{*}$ is _simple_ if each of its
characters appears exactly once. Two words in $[{{}\mathbf{n}}_{\bullet}]^{*}$
are _disjoint_ if they share no common character. We denote by $\ell_{w}$ the
length of $w\in[{{}\mathbf{n}}_{\bullet}]^{*}$. Further set

$\displaystyle\mathcal{X}\operatorname{\coloneqq}$ $\displaystyle\
\left\\{\left(w_{1},\dotsc,w_{r}\right):r\in{\mathbb{N}}_{1}\,\mathrm{,}\;\,w_{i}\in[{{}\mathbf{n}}_{\bullet}]^{*}\,\mathrm{,}\;\,w_{i}\text{
simple, pairwise disjoint}\,\mathrm{,}\;\,\
\sum_{i=1}^{r}\ell_{w_{i}}={{}\mathbf{n}}_{\bullet}\right\\}\,\mathrm{,}\;\,$
$\displaystyle\mathcal{Z}\operatorname{\coloneqq}$ $\displaystyle\
\big{\\{}\left\\{w_{1},\dotsc,w_{r}\right\\}:\left(w_{1},\dotsc,w_{r}\right)\in\mathcal{X}\big{\\}}\,\mathrm{,}\;\,$

and note that, since the necklace $\llbracket w\rrbracket$ of a simple word
$w\in[{{}\mathbf{n}}_{\bullet}]^{*}$ is a cycle in
$\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$, then

$\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}=\big{\\{}\llbracket
w_{1}\rrbracket\cdots\llbracket
w_{r}\rrbracket:\left\\{w_{1},\dotsc,w_{r}\right\\}\in\mathcal{Z}\big{\\}}\,\mathrm{.}$

  2. $(b)$

Let
$\mathfrak{c}_{\mathbf{n}}^{*}\colon[{{}\mathbf{n}}_{\bullet}]^{*}\to[q]^{*}$
be defined by $\mathfrak{c}_{\mathbf{n}}^{*}\colon
w\operatorname{\coloneqq}y_{1}\cdots
y_{\ell}\longmapsto\mathfrak{c}_{\mathbf{n}}(y_{1})\cdots\mathfrak{c}_{\mathbf{n}}(y_{\ell})$,
and denote again by $\mathfrak{c}_{\mathbf{n}}^{*}$ its component-wise
extension to $\mathcal{X}$.

  3. $(c)$

Set
$\mathcal{V}\operatorname{\coloneqq}\mathfrak{c}_{\mathbf{n}}^{*}(\mathcal{X})$,
denote again by $\llbracket{\,\cdot\,}\rrbracket$ the component-wise extension
to $\mathcal{V}$ of the quotient map $\llbracket{\,\cdot\,}\rrbracket$ from
$[q]^{*}$ to necklaces, and set
$\mathcal{U}\operatorname{\coloneqq}\left\\{{\big{(}\llbracket
v_{1}\rrbracket,\dotsc,\llbracket
v_{r}\rrbracket\big{)}}:\left(v_{1},\dotsc,v_{r}\right)\in\mathcal{V}\right\\}$.

  4. $(d)$

Define a map $\boldsymbol{\nu}$ on $\mathcal{U}$ by
$\boldsymbol{\nu}\colon{\big{(}\llbracket v_{1}\rrbracket,\dotsc,\llbracket
v_{r}\rrbracket\big{)}}\longmapsto\sum_{i=1}^{r}\operatorname{\mathbf{1}}_{\llbracket
v_{i}\rrbracket}$.

  5. $(e)$

Finally, define maps $f\colon\mathcal{X}\to\mathcal{Z}$ and
$\llbracket{\,\cdot\,}\rrbracket^{*}\colon\mathcal{Z}\to\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$
by

$f\colon\left(w_{1},\dotsc,w_{r}\right)\longmapsto\left\\{w_{1},\dotsc,w_{r}\right\\}\,\mathrm{,}\;\,\qquad\llbracket{\,\cdot\,}\rrbracket^{*}\colon\left\\{w_{1},\dotsc,w_{r}\right\\}\longmapsto\llbracket
w_{1}\rrbracket\cdots\llbracket w_{r}\rrbracket\,\mathrm{.}$

${\mathcal{X}}$${\mathcal{Z}}$${\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}}$${\mathcal{V}}$${\mathcal{U}}$${\mathcal{N}_{\mathbf{n}}}$${\mathcal{A}_{\mathbf{n}}}$$\scriptstyle{\mathfrak{c}_{\mathbf{n}}^{*}}$$\scriptstyle{f}$$\scriptstyle{\llbracket{\,\cdot\,}\rrbracket^{*}}$$\scriptstyle{\mathfrak{c}^{\diamond}_{\mathbf{n}}}$$\scriptstyle{\Pi}$$\scriptstyle{\llbracket{\,\cdot\,}\rrbracket}$$\scriptstyle{\boldsymbol{\nu}}$$\scriptstyle{\boldsymbol{\varepsilon}_{*}}$
Figure 2: Auxiliary maps and sets in the proof of Proposition 2.15.

It is a tedious verification that the diagram in Figure 2 commutes, and a
simple computation of the cardinality of the fibers of the maps involved
yields the conclusion. ∎

##  3 Multivariate moments

For $k\geq 1$ let $\Delta^{k-1}$ be the _standard simplex_

$\Delta^{k-1}\operatorname{\coloneqq}\big{\\{}\mathbf{x}\in{\mathbb{R}}^{k}:x_{i}\geq
0\,\mathrm{,}\;\,x_{1}+\cdots+x_{k}=1\big{\\}}\,\mathrm{,}\;\,$ (3.1)

and recall the definition (1) of the Dirichlet distribution
$D_{\boldsymbol{\alpha}}$.

Our main result in this section is a formula for the multivariate moments of
$D_{\boldsymbol{\alpha}}$.

######  Theorem 3.1 (Multivariate moments of $D_{\boldsymbol{\alpha}}$).

The following identity holds

$\mu_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\operatorname{\coloneqq}\int_{\Delta^{k-1}}\prod_{j}^{q}(\mathbf{s}_{j}\cdot\mathbf{x})^{n_{j}}\mathop{}\\!\mathrm{d}D_{\boldsymbol{\alpha}}(\mathbf{x})=\frac{\mathbf{n}!}{\left\langle{{}\boldsymbol{\alpha}}_{\bullet}\right\rangle_{{{}\mathbf{n}}_{\bullet}}}\,Z_{\mathbf{n}}\big{(}{\Omega_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]}\big{)}\eqqcolon\zeta_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\,\mathrm{.}$
(3.2)

In order to put Theorem 3.1 into context, we briefly survey previously known
results on moments of Dirichlet and related measures.

###  3.1 Overview on Dirichlet measures

Moment and Laplace/Fourier-transform methods for $D_{\boldsymbol{\alpha}}$ and
its infinite-dimensional counterpart, the Dirichlet–Ferguson measure
$\mathcal{D}_{\alpha}$ [12] over a measure space $(X,\alpha)$ are notoriously
difficult, as we briefly summarize below.

##### Transforms

It is well-known that the Fourier transform
$\widehat{D_{\boldsymbol{\alpha}}}$ of the Dirichlet distribution
$D_{\boldsymbol{\alpha}}$ may be expressed in terms of ${}_{k}\Phi_{2}$, the
$k$-variate _confluent hypergeometric Lauricella function of type $D$_ [20,
10]. The power-series representation of ${}_{k}\Phi_{2}$ is « _inconvenient
for numerical calculations when $k>2$_» [22, p. 4]. Instead, the complex-
contour-integral representation [7, Eqn. (7)] is preferred, but its treatment
remains quite involved, see e.g. [25]. In particular, differentiating
$\widehat{D_{\boldsymbol{\alpha}}}$ in this form does not provide any useful
representation for the moments of the measure.

For decades the Fourier transform $\widehat{\mathcal{D}_{\alpha}}$ of
$\mathcal{D}_{\alpha}$ was widely considered intractable [14], which led to
the introduction of other characterizing transforms, such as the _Markov–Krein
transform_ [15] and the _$c$ -transform_ [14]. These methods too are
unsatisfactory, since there is no counterpart for such transforms of
foundational results available for the Fourier transform, such as, for
instance, Bochner–Minlos–Sazonov’s (BMS) or Lévy’s Continuity Theorem. The
Fourier transform $\widehat{\mathcal{D}_{\alpha}}$ was eventually computed in
[4] by methods in combinatorics and representation theory.

##### Moments

Multivariate moments of $D_{\boldsymbol{\alpha}}$ are virtually well-known in
the form (3.5) below, which may be regarded as an extension of the ESF.
Whereas easily computable for small $k$, this form is unsuitable for letting
$k\to\infty$ and thus provides no insight on multivariate moments of
$\mathcal{D}_{\alpha}$.

Partially in order to overcome this issue, other expressions have been
considered:  $(a)$ the univariate moment $\mathcal{D}_{\alpha}(f^{n})$ has
appeared in [24] in terms of incomplete Bell polynomials, solely in the case
$X\Subset{\mathbb{R}}$ and $f=\mathrm{id}_{\mathbb{R}}$;  $(b)$ more general
univariate moments for $D_{\boldsymbol{\alpha}}$ have implicitly appeared in
[21, proof of Prop. 3.3] in iterative form;  $(c)$ univariate moments for both
$D_{\boldsymbol{\alpha}}$ and $\mathcal{D}_{\alpha}$ have appeared in full
generality in [4] in terms of the cycle index polynomials $Z_{n}$, which
allowed the aforementioned computation of $\widehat{\mathcal{D}_{\alpha}}$. As
for multivariate moments, they have appeared:  $(d)$ in [15, Prop. 7.4], in
terms of summations over constrained permutations, only in the case
${{}\boldsymbol{\alpha}}_{\bullet}=1$;  $(e)$ in [8, Eqn. (4.20)], [11, Lem.
5.2], and [6, Cor. 3.5], in terms of summations over constrained _set_
partitions.

##### Other measures

The measure $\mathcal{D}_{\alpha}$ is the _simplicial part_ of other known
measures on the space $\mathscr{M}^{+}$ of non-negative Borel measures on $X$.
Among them are: the law $\mathcal{G}_{\alpha}$ of the _$\gamma$ -point
process_ [18] with intensity $\alpha$, and A.M. Vershik’s _multiplicative
infinite-dimensional Lebesgue measure_ $\mathcal{L}_{\alpha}$ [30, 31] with
intensity $\alpha$. Together with $\mathcal{D}_{\alpha}$, these measures have
a wide range of applications, from the theory of point processes and of
measure-valued Markov diffusions, see [5, §1] and references therein, to the
representation theory of infinite-dimensional Lie groups of
currents/multipliers, see [29], or [4, §1] for a unified treatment.

In §3.3 we give moment formulas for $\mathcal{D}_{\alpha}$ and
$\mathcal{G}_{\alpha}$ analog to the one in Theorem 3.1.

##### Relations to the ESF

One relation between the Dirichlet distribution and the ESF is made apparent
by the expression of the generating function of $E_{\theta}$ in the dummy
variables $\mathbf{t}\operatorname{\coloneqq}\left(t_{1},\dotsc,t_{n}\right)$
in terms of the cycle index polynomial $Z_{n}$ (2.4) of $\mathfrak{S}_{n}$,
viz.

$\sum_{\boldsymbol{\lambda}\vdash
n}E_{\theta}(\boldsymbol{\lambda})\,\mathbf{t}^{\boldsymbol{\lambda}}=\frac{n!}{\left\langle\theta\right\rangle_{n}}Z_{n}[\theta\,\mathbf{t}]\,\mathrm{,}\;\,\qquad\mathbf{t}^{\boldsymbol{\lambda}}\operatorname{\coloneqq}t_{1}^{\lambda_{1}}\cdots
t_{n}^{\lambda_{n}}\,\mathrm{.}$ (3.3)

###  3.2 Some corollaries

Let us collect some corollaries and special cases of Theorem 3.1.

######  Corollary 3.2.

Let $P_{\pi}$ be the permutation matrix of a permutation
$\pi\in\mathfrak{S}_{q}$. Then,

$Z_{\mathbf{n}}[\boldsymbol{\alpha};\mathbf{S}]=Z_{P_{\pi}\mathbf{n}}[\boldsymbol{\alpha};\mathbf{S}P_{\pi}]\,\mathrm{,}\;\,\qquad\mathbf{S}\in{\mathbb{R}}^{k\times
q}\,\mathrm{.}$

######  Corollary 3.3.

For every $\mathbf{n}\in{\mathbb{N}}^{q}_{*}$ we have

$\sum_{\begin{subarray}{c}A\vDash\mathbf{n}\\\
\mathsf{shape}(A)=\boldsymbol{\lambda}\end{subarray}}M_{\mathbf{n}}(A)=M_{2}(\boldsymbol{\lambda})\,\mathrm{.}$

###### Proof.

In (3.2), choose $\mathbf{s}_{1}=\dotsc=\mathbf{s}_{q}\eqqcolon\mathbf{s}$ and
$\boldsymbol{\alpha}$ with ${{}\boldsymbol{\alpha}}_{\bullet}=1$, and set
$n\operatorname{\coloneqq}{{}\mathbf{n}}_{\bullet}$. Then, the left-hand side
of (3.2) becomes the $n^{\textrm{th}}$-moment of the linear functional
$\mathbf{x}\mapsto\mathbf{s}\cdot\mathbf{x}$ of $D_{\boldsymbol{\alpha}}$ and
is thus equal to $Z_{n}[\mathbf{s}^{\diamond
1}\cdot\boldsymbol{\alpha},\dotsc,\mathbf{s}^{\diamond
n}\cdot\boldsymbol{\alpha}]$ by [4, Thm. 3.2]. As for the right-hand side, for
the above choice of the $\mathbf{s}_{i}$’s the monomials
$\omega_{\mathbf{a}}[\mathbf{S};\boldsymbol{\alpha}]$ satisfy
$\omega_{\mathbf{a}}[\mathbf{S};\boldsymbol{\alpha}]=\omega_{\mathbf{a}^{\prime}}[\mathbf{S};\boldsymbol{\alpha}]$
whenever ${{}\mathbf{a}}_{\bullet}={{}\mathbf{a}^{\prime}}_{\bullet}$.
Collecting terms in the right-hand side and equating the coefficients of the
corresponding monomials on both sides yields the assertion. ∎

######  Corollary 3.4.

The following identity holds

$\sum_{\begin{subarray}{c}\mathbf{n}\in{\mathbb{N}}_{0}^{q}\\\
{{}\mathbf{n}}_{\bullet}=n\end{subarray}}Z_{\mathbf{n}}\big{(}{\Omega_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]}\big{)}=Z_{n}[\mathsf{row}(\mathbf{S})\cdot\boldsymbol{\alpha},\mathsf{row}(\mathbf{S})^{\diamond
2}\cdot\boldsymbol{\alpha},\dotsc,\mathsf{row}(\mathbf{S})^{\diamond
n}\cdot\boldsymbol{\alpha}]\,\mathrm{.}$ (3.4)

###### Proof.

Set

$\displaystyle\Phi[\boldsymbol{\alpha};\mathbf{S}]\operatorname{\coloneqq}\sum_{\mathbf{M}\in{\mathbb{N}}_{0}^{k\times
q}}\frac{\left\langle\boldsymbol{\alpha}\right\rangle_{\mathsf{row}(\mathbf{M})}}{\left\langle{{}\boldsymbol{\alpha}}_{\bullet}\right\rangle_{{{}\mathsf{col}(\mathbf{M})}_{\bullet}}}\frac{\mathbf{S}^{\mathbf{M}}}{\mathbf{M}!}\,\mathrm{.}$

By definition of $\Phi[\boldsymbol{\alpha};\mathbf{S}]$ and Lemma 3.8 below,
and by [4, Eqn. (2.9)], for every $t\in{\mathbb{R}}$,

$\Phi[\boldsymbol{\alpha};t\,\mathbf{S}]=\int_{\Delta^{k-1}}e^{t(\mathbf{s}_{1}+\cdots+\mathbf{s}_{q})\cdot\mathbf{x}}\mathop{}\\!\mathrm{d}D_{\boldsymbol{\alpha}}(\mathbf{x})\eqqcolon\widehat{D_{\boldsymbol{\alpha}}}\big{(}{\mathsf{row}(\mathbf{S})}\big{)}={}_{k}\Phi_{2}[\boldsymbol{\alpha};{{}\boldsymbol{\alpha}}_{\bullet};t\,\mathsf{row}(\mathbf{S})]\,\mathrm{.}$

Expanding the left-hand side as a series in $n\in{\mathbb{N}}_{0}$, each
summand is the left-hand side of (3.4) by Theorem 3.1. Expanding the right-
hand side as series in $n\in{\mathbb{N}}_{0}$, each summand is the right-hand
side of (3.4) by [4, Prop. 3.5]. Since, for _same_ $n$, the summands in each
of these expansions are polynomials of _same_ degree equal to $n$ in the
variables $\mathbf{S}$, we may equate the summands one by one, which yields
(3.4). ∎

###  3.3 Dirichlet–Ferguson and Gamma measures

Let $X$ be a second countable locally compact Hausdorff space, and
$\mathscr{P}$ be the space of all Borel probability measures on $X$, endowed
with the Borel $\sigma$-algebra of the narrow topology. For any finite Borel
measure $\eta$ on $X$ and any bounded Borel $f\colon X\to{\mathbb{R}}$ we set
$\eta f\operatorname{\coloneqq}\int f\mathop{}\\!\mathrm{d}\eta$.

##### Dirichlet–Ferguson measures

For $\beta>0$ and $\sigma\in\mathscr{P}$, let
$\alpha\operatorname{\coloneqq}\beta\sigma$ be the finite Borel measure on $X$
with total mass $\beta$ and _shape_ (also: _simplicial part_) $\sigma$. The
_Dirichlet–Ferguson measure $\mathcal{D}_{\alpha}$ with intensity_ (_measure_)
$\alpha$ is the unique Borel probability measure on $\mathscr{P}$ with Fourier
transform [4, Thm. 3.10]

$\displaystyle\widehat{\mathcal{D}_{\alpha}}(f)\operatorname{\coloneqq}\int_{\mathscr{P}}e^{\mathbbm{i}\,\eta
f}\mathop{}\\!\mathrm{d}\mathcal{D}_{\alpha}(f)=\sum_{n=0}^{\infty}\frac{\mathbbm{i}^{n}}{\left\langle\beta\right\rangle_{n}}Z_{n}\big{(}{\alpha
f,\alpha f^{2},\dotsc,\alpha f^{n}}\big{)}\,\mathrm{,}\;\,\qquad
f\in\mathcal{C}_{b}\,\mathrm{.}$

For continuous bounded $f_{1},\dotsc,f_{q}\colon X\to{\mathbb{R}}$, set

$\displaystyle\Omega_{\mathbf{n}}[f_{1},\dotsc,f_{q};\alpha]\operatorname{\coloneqq}$
$\displaystyle\ \left(\alpha\big{(}{f_{1}^{h_{1}}\cdots
f_{q}^{h_{q}}}\big{)}\right)_{\mathbf{h}\leq_{\diamond}\mathbf{n}}\,\mathrm{.}$

By a straightforward adaptation of the proof for the univariate case [4, Thm.
3.10], as a corollary of Theorem 3.1 we obtain an explicit expression for the
moments of $\mathcal{D}_{\alpha}$.

######  Corollary 3.5 (Multivariate moments of $\mathcal{D}_{\alpha}$).

We have

$\int_{\mathscr{P}}\prod_{j}^{q}(\eta
f_{j})^{n_{j}}\mathop{}\\!\mathrm{d}\mathcal{D}_{\alpha}(\eta)=\frac{\mathbf{n}!}{\left\langle\beta\right\rangle_{{{}\mathbf{n}}_{\bullet}}}Z_{\mathbf{n}}\big{(}{\Omega_{\mathbf{n}}[f_{1},\dotsc,f_{q};\alpha]}\big{)}\,\mathrm{.}$

We recover Theorem 3.1 by choosing a Borel partition
$\left(X_{i}\right)_{i}^{k}$ of $X$ with
$\alpha_{i}\operatorname{\coloneqq}\alpha X_{i}$, and simple functions
$f_{1},\dotsc,f_{q}$, constantly equal to, respectively,
$s_{1,i},\dotsc,s_{q,i}$ on each set $X_{i}$ for each $i\in[k]$.

##### Gamma measures

Let $\mathcal{G}_{\alpha}$ be the law of the Gamma point process with
intensity $\alpha$, e.g. [18].

######  Corollary 3.6 (Multivariate moments of $\mathcal{G}_{\alpha}$).

We have

$\int_{\mathscr{M}_{b}^{+}}\prod_{j}^{q}(\eta
f_{j})^{n_{j}}\mathop{}\\!\mathrm{d}\mathcal{G}_{\alpha}(\eta)=\mathbf{n}!\,Z_{\mathbf{n}}\big{(}{\Omega_{\mathbf{n}}[f_{1},\dotsc,f_{q};\alpha]}\big{)}\,\mathrm{.}$

######  Remark 3.7.

Alternative expressions for the multivariate moments of the Gamma measure may
be obtained by differentiating its characteristic functional (e.g. [6, p. 5]).
Such expressions are however not informative on their algebraic and
combinatorial meaning in connection with $Z_{\mathbf{n}}$, as they rather rely
on the multivariate multi-factor Leibniz rule. A similar approach does not
apply to the Dirichlet–Ferguson measure, due to the convoluted form of its
characteristic functional.

###  3.4 Proof of Theorem 3.1

######  Lemma 3.8.

The following identity holds

$\mu_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=\frac{\mathbf{n}!}{\left\langle{{}\boldsymbol{\alpha}}_{\bullet}\right\rangle_{{{}\mathbf{n}}_{\bullet}}}\sum_{\begin{subarray}{c}\mathbf{M}\in{\mathbb{N}}_{0}^{k\times
q}\\\
\mathsf{col}(\mathbf{M})=\mathbf{n}\end{subarray}}\left\langle\boldsymbol{\alpha}\right\rangle_{\mathsf{row}(\mathbf{M})}\frac{\mathbf{S}^{\mathbf{M}}}{\mathbf{M}!}\eqqcolon\nu_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\,\mathrm{.}$
(3.5)

###### Proof.

By the Multinomial Theorem and by properties of the Dirichlet distribution

$\displaystyle\mu_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=$
$\displaystyle\
\frac{1}{\mathrm{B}[\boldsymbol{\alpha}]}\int_{\Delta^{k-1}}\left(\prod_{j}^{q}\sum_{\begin{subarray}{c}\mathbf{m}\in{\mathbb{N}}_{0}^{k}\\\
{{}\mathbf{m}}_{\bullet}=n_{j}\end{subarray}}\binom{n_{j}}{\mathbf{m}}\,\mathbf{s}_{j}^{\mathbf{m}}\,\mathbf{x}^{\mathbf{m}}\right)\mathbf{x}^{\boldsymbol{\alpha}-{\mathbf{1}}}\mathop{}\\!\mathrm{d}\mathbf{x}$
$\displaystyle=$ $\displaystyle\
\frac{1}{\mathrm{B}[\boldsymbol{\alpha}]}\int_{\Delta^{k-1}}\left(\sum_{\begin{subarray}{c}\mathbf{m}_{1},\dotsc,\mathbf{m}_{q}\in{\mathbb{N}}_{0}^{k}\\\
{{}\mathbf{m}_{1}}_{\bullet}=n_{1},\dotsc,{{}\mathbf{m}_{q}}_{\bullet}=n_{q}\end{subarray}}\prod_{j}^{q}\binom{n_{j}}{\mathbf{m}_{j}}\,\mathbf{s}_{j}^{\mathbf{m}_{j}}\mathbf{x}^{\mathbf{m}_{j}}\right)\mathbf{x}^{\boldsymbol{\alpha}-{\mathbf{1}}}\mathop{}\\!\mathrm{d}\mathbf{x}$
$\displaystyle=$ $\displaystyle\
\sum_{\begin{subarray}{c}\mathbf{m}_{1},\dotsc,\mathbf{m}_{q}\in{\mathbb{N}}_{0}^{k}\\\
{{}\mathbf{m}_{1}}_{\bullet}=n_{1},\dotsc,{{}\mathbf{m}_{q}}_{\bullet}=n_{q}\end{subarray}}\frac{1}{\mathrm{B}[\boldsymbol{\alpha}]}\left(\prod_{j}^{q}\binom{n_{j}}{\mathbf{m}_{j}}\,\mathbf{s}_{j}^{\mathbf{m}_{j}}\right)\int_{\Delta^{k-1}}\mathbf{x}^{\mathbf{m}_{1}+\cdots+\mathbf{m}_{q}+\boldsymbol{\alpha}-{\mathbf{1}}}\mathop{}\\!\mathrm{d}\mathbf{x}$
$\displaystyle=$ $\displaystyle\
\sum_{\begin{subarray}{c}\mathbf{m}_{1},\dotsc,\mathbf{m}_{q}\in{\mathbb{N}}_{0}^{k}\\\
{{}\mathbf{m}_{1}}_{\bullet}=n_{1},\dotsc,{{}\mathbf{m}_{q}}_{\bullet}=n_{q}\end{subarray}}\frac{\mathrm{B}[\mathbf{m}_{1}+\cdots+\mathbf{m}_{q}+\boldsymbol{\alpha}]}{\mathrm{B}[\boldsymbol{\alpha}]}\prod_{j}^{q}\binom{n_{j}}{\mathbf{m}_{j}}\,\mathbf{s}_{j}^{\mathbf{m}_{j}}\,\mathrm{.}$

Reindexing the summation over
$\mathbf{M}=(\mathbf{m}_{1},\dotsc,\mathbf{m}_{q})\in{\mathbb{R}}^{k\times
q}$, we conclude that

$\displaystyle\mu_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=$
$\displaystyle\
\frac{\mathbf{n}!}{\left\langle{{}\boldsymbol{\alpha}}_{\bullet}\right\rangle_{{{}\mathbf{n}}_{\bullet}}}\sum_{\begin{subarray}{c}\mathbf{M}\in{\mathbb{N}}_{0}^{q\times
k}\\\
\mathsf{col}(\mathbf{M})=\mathbf{n}\end{subarray}}\left\langle\boldsymbol{\alpha}\right\rangle_{\mathsf{row}(\mathbf{M})}\frac{\mathbf{S}^{\mathbf{M}}}{\mathbf{M}!}\,\mathrm{.}\qed$

Let us recall the following fact, e.g. [27, Eqn. (8), p. 1060], or [2, Eqn.
(10), p. 39].

######  Lemma 3.9.

For every $k\in{\mathbb{N}}$, every $\mathbf{v}\in{\mathbb{N}}_{0}^{k}$, and
every integer $0\leq m\leq{{}\mathbf{v}}_{\bullet}$,

$\binom{{{}\mathbf{v}}_{\bullet}}{\mathbf{v}}=\sum_{\begin{subarray}{c}\mathbf{w}\in{\mathbb{N}}_{0}^{k}\\\
\mathbf{w}\leq_{\diamond}\mathbf{v}\\\
{{}\mathbf{w}}_{\bullet}=m\end{subarray}}\binom{{{}\mathbf{w}}_{\bullet}}{\mathbf{w}}\binom{{{}\mathbf{v}}_{\bullet}-{{}\mathbf{w}}_{\bullet}}{\mathbf{v}-\mathbf{w}}\,\mathrm{.}$
(3.6)

###### Proof of Theorem 3.1.

Set

$\tilde{\nu}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\operatorname{\coloneqq}\frac{\left\langle\boldsymbol{\alpha}\right\rangle_{{{}\mathbf{n}}_{\bullet}}}{\mathbf{n}!}\nu_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\qquad\text{and}\qquad\tilde{\zeta}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\operatorname{\coloneqq}\frac{\left\langle\boldsymbol{\alpha}\right\rangle_{{{}\mathbf{n}}_{\bullet}}}{\mathbf{n}!}\zeta_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\,\mathrm{.}$

We show that
$\tilde{\nu}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=\tilde{\zeta}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]$
and conclude the assertion by Lemma 3.8.

##### Step 1

We claim that

$\displaystyle\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{j}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{\ell}]=\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\,\tilde{\nu}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]\,\mathrm{,}\;\,$
(3.7)

where, conventionally,

$\tilde{\nu}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=0\quad\text{whenever}\quad\mathbf{n}\not\geq^{\diamond}\operatorname{{\mathbf{0}}}_{q}\,\mathrm{.}$
(3.8)

We argue by induction on ${{}\mathbf{n}}_{\bullet}$ with trivial (i.e. $1=1$)
base step for ${{}\mathbf{n}}_{\bullet}=1$.

_Inductive step._ Let
$\partial_{a}^{b}\operatorname{\coloneqq}\partial_{s_{a}^{b}}$, set
$\mathbf{E}_{a}^{b}\operatorname{\coloneqq}[\delta_{ai}\delta_{bj}]_{i}^{j}\in\left\\{0,1\right\\}^{k\times
q}$, and note that

$\displaystyle\partial_{a}^{b}\tilde{\nu}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=$
$\displaystyle\
\sum_{\begin{subarray}{c}\mathbf{M}\in{\mathbb{N}}_{0}^{k\times q}\\\
\mathsf{col}(\mathbf{M})=\mathbf{n}\end{subarray}}\left\langle\boldsymbol{\alpha}\right\rangle_{\mathsf{row}(\mathbf{M})}\partial_{a}^{b}\frac{\mathbf{S}^{\mathbf{M}}}{\mathbf{M}!}=\sum_{\begin{subarray}{c}\mathbf{E}_{a}^{b}\leq_{\diamond}\mathbf{M}\in{\mathbb{N}}_{0}^{k\times
q}\\\
\mathsf{col}(\mathbf{M})=\mathbf{n}\end{subarray}}\alpha_{a}\left\langle\boldsymbol{\alpha}+\mathbf{e}_{a}\right\rangle_{\mathsf{row}(\mathbf{M}-\mathbf{E}_{a}^{b})}\frac{\mathbf{S}^{\mathbf{M}-\mathbf{E}_{a}^{b}}}{(\mathbf{M}-\mathbf{E}_{a}^{b})!}$
$\displaystyle=$ $\displaystyle\
\sum_{\begin{subarray}{c}\mathbf{M}\in{\mathbb{N}}_{0}^{k\times q}\\\
\mathsf{col}(\mathbf{M})=\mathbf{n}-\mathbf{e}_{b}\end{subarray}}\alpha_{a}\left\langle\boldsymbol{\alpha}+\mathbf{e}_{a}\right\rangle_{\mathsf{row}(\mathbf{M})}\frac{\mathbf{S}^{\mathbf{M}}}{\mathbf{M}!}$
$\displaystyle=$ $\displaystyle\
\alpha_{a}\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{b}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{a}]\,\mathrm{.}$
(3.9)

Applying the inductive hypothesis to $\mathbf{n}-\mathbf{e}_{b}$ with
$\boldsymbol{\alpha}+\mathbf{e}_{a}$ in place of $\boldsymbol{\alpha}$, we
have

$\displaystyle\alpha_{a}\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{j}-\mathbf{e}_{b}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{\ell}+\mathbf{e}_{a}]=$
$\displaystyle\
\alpha_{a}\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}-\mathbf{e}_{b}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\,\tilde{\nu}_{\mathbf{n}-\mathbf{h}-\mathbf{e}_{b}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{a}]$
$\displaystyle=$ $\displaystyle\
\alpha_{a}\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\,\tilde{\nu}_{\mathbf{n}-\mathbf{h}-\mathbf{e}_{b}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{a}]\,\mathrm{,}\;\,$
(3.10)

where the last equality holds by (3.8).

Now, let $\ell\neq a$. Applying (3.9) with $\mathbf{n}-\mathbf{e}_{j}$ in
place of $\mathbf{n}$ and $\boldsymbol{\alpha}+\mathbf{e}_{\ell}$ in place of
$\boldsymbol{\alpha}$,

$\displaystyle\partial_{a}^{b}\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{j}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{\ell}]=\alpha_{a}\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{j}-\mathbf{e}_{b}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{\ell}+\mathbf{e}_{a}]\,\mathrm{.}$
(3.11)

Since $\ell\neq a$, applying (3.9) with $\mathbf{n}-\mathbf{h}$ in place of
$\mathbf{n}$ for every $\mathbf{h}\leq_{\diamond}\mathbf{n}$ yields

$\displaystyle\partial_{a}^{b}$
$\displaystyle\left(\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\tilde{\nu}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]\right)=$
(3.12) $\displaystyle\
=\alpha_{a}\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\,\tilde{\nu}_{\mathbf{n}-\mathbf{h}-\mathbf{e}_{b}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{a}]\,\mathrm{.}$

Combining (3.11), (3.12), and (3.10) yields

$\displaystyle\partial_{a}^{b}\left(\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{j}}[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{\ell}]-\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\tilde{\nu}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]\right)=0\,\mathrm{.}$

By arbitrariness of $a,b$ we conclude that the bracketed quantity is a
polynomial in the sole variables $s_{\ell}^{1},\dotsc,s_{\ell}^{q}$. As a
consequence, every monomial not in the sole variables
$s_{\ell}^{1},\dotsc,s_{\ell}^{q}$ cancels out by arbitrariness of the
variables $\mathbf{S}$. Thus,

$\displaystyle\tilde{\nu}_{\mathbf{n}-\mathbf{e}_{j}}$
$\displaystyle[\mathbf{S};\boldsymbol{\alpha}+\mathbf{e}_{\ell}]-\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\tilde{\nu}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]$
$\displaystyle=\left\langle\alpha_{\ell}+1\right\rangle_{{{}\mathbf{n}}_{\bullet}-1}\frac{\mathbf{S}_{\ell}^{\mathbf{n}-\mathbf{e}_{j}}}{(\mathbf{n}-\mathbf{e}_{j})!}-\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\mathbf{S}_{\ell}^{\mathbf{h}-\mathbf{e}_{j}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\left\langle\alpha_{\ell}\right\rangle_{{{}\mathbf{n}}_{\bullet}-{{}\mathbf{h}}_{\bullet}}\frac{\mathbf{S}_{\ell}^{\mathbf{n}-\mathbf{h}}}{(\mathbf{n}-\mathbf{h})!}\,\mathrm{.}$

The latter quantity is proved to vanish as soon as

$\displaystyle\frac{\left\langle\alpha_{\ell}+1\right\rangle_{{{}\mathbf{n}}_{\bullet}-1}}{(\mathbf{n}-\mathbf{e}_{j})!}=\sum_{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\frac{\left\langle\alpha_{\ell}\right\rangle_{{{}\mathbf{n}}_{\bullet}-{{}\mathbf{h}}_{\bullet}}}{(\mathbf{n}-\mathbf{h})!}\,\mathrm{,}\;\,$

or, equivalently, by the Chu–Vandermonde identity,

$\displaystyle\frac{({{}\mathbf{n}}_{\bullet}-1)!}{(\mathbf{n}-\mathbf{e}_{j})!}\sum_{i=1}^{{{}\mathbf{n}}_{\bullet}}\frac{\left\langle\alpha_{\ell}\right\rangle_{{{}\mathbf{n}}_{\bullet}-i}}{({{}\mathbf{n}}_{\bullet}-i)!}=\sum_{i=1}^{{{}\mathbf{n}}_{\bullet}}\sum_{\begin{subarray}{c}\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}\\\
{{}\mathbf{h}}_{\bullet}=i\end{subarray}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\frac{\left\langle\alpha_{\ell}\right\rangle_{{{}\mathbf{n}}_{\bullet}-i}}{(\mathbf{n}-\mathbf{h})!}\,\mathrm{.}$

The latter is implied by the equality of each of the summands, viz.

$\displaystyle\frac{({{}\mathbf{n}}_{\bullet}-1)!}{(\mathbf{n}-\mathbf{e}_{j})!}\frac{1}{({{}\mathbf{n}}_{\bullet}-i)!}=\sum_{\begin{subarray}{c}{\mathbf{e}_{j}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\\\
{{}\mathbf{h}}_{\bullet}=i\end{subarray}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{j})!}\frac{1}{(\mathbf{n}-\mathbf{h})!}\,\mathrm{,}\;\,$

which is in turn a consequence of Lemma 3.9, after relabeling $\mathbf{n}$ as
$\mathbf{n}-\mathbf{e}_{j}$.

##### Step 2

We now verify that $\tilde{\nu}_{\mathbf{n}}=\tilde{\zeta}_{\mathbf{n}}$. We
argue by strong induction on ${{}\mathbf{n}}_{\bullet}$ with trivial (i.e.
$1=1$) base step ${{}\mathbf{n}}_{\bullet}=0$. _Inductive step._ Assume for
every $\boldsymbol{\alpha}\in{\mathbb{R}}^{k}_{+}$ that
$\tilde{\nu}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]=\tilde{\zeta}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]$
for every $\mathbf{h}\leq_{\diamond}\mathbf{n}$ with
$\mathbf{h}\neq\operatorname{{\mathbf{0}}}$. Now,

$\displaystyle\mathbf{n}!\,\partial_{p}^{b}\tilde{\zeta}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=$
$\displaystyle\
\sum_{A\vDash\mathbf{n}}M_{\mathbf{n}}(A)\,\partial_{p}^{b}\underbrace{\prod_{\mathbf{a}\in\mathsf{supp}A}\left(\big{(}{\mathbf{s}_{1}^{\diamond
a_{1}}\diamond\cdots\diamond\mathbf{s}_{q}^{\diamond
a_{q}}}\big{)}\cdot\boldsymbol{\alpha}\right)^{A(\mathbf{a})}}_{\eqqcolon
J(\mathbf{S};\boldsymbol{\alpha};\mathbf{A})}$ $\displaystyle=$
$\displaystyle\
\sum_{A\vDash\mathbf{n}}M_{\mathbf{n}}(A)\,\sum_{\begin{subarray}{c}\mathbf{a}\in\mathsf{supp}A\\\
\mathbf{a}\geq_{\diamond}\mathbf{e}_{b}\end{subarray}}\frac{A(\mathbf{a})\,a_{b}\,\alpha_{p}\,\mathbf{S}_{p}^{\mathbf{a}-\mathbf{e}_{b}}}{\big{(}{\mathbf{s}_{1}^{\diamond
a_{1}}\diamond\cdots\diamond\mathbf{s}_{q}^{\diamond
a_{q}}}\big{)}\cdot\boldsymbol{\alpha}}\,J(\mathbf{S};\boldsymbol{\alpha};A)$
$\displaystyle=$ $\displaystyle\
\alpha_{p}\sum_{\mathbf{e}_{b}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\sum_{A\vDash\mathbf{n}}M_{\mathbf{n}}(A)\frac{A(\mathbf{h})\,h_{b}\,\mathbf{S}_{p}^{\mathbf{h}-\mathbf{e}_{b}}}{\big{(}{\mathbf{s}_{1}^{\diamond
h_{1}}\diamond\cdots\diamond\mathbf{s}_{q}^{\diamond
h_{q}}}\big{)}\cdot\boldsymbol{\alpha}}\,J(\mathbf{S};\boldsymbol{\alpha};A)\,\mathrm{.}$
(3.13)

For each $\mathbf{h}\leq_{\diamond}\mathbf{n}$ and each $A\vDash\mathbf{n}$
with $A\geq\operatorname{\mathbf{1}}_{\mathbf{h}}$, set
$C\operatorname{\coloneqq}A-\operatorname{\mathbf{1}}_{\mathbf{h}}$. Note that

$\displaystyle M_{\mathbf{n}}(A)=$ $\displaystyle\
\mathbf{n}!\prod_{\mathbf{a}\in\mathsf{supp}A}\frac{\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}}{{{}\mathbf{a}}_{\bullet}^{A(\mathbf{a})}A(\mathbf{a})!}=\frac{\mathbf{n}!\,{{}\mathbf{h}}_{\bullet}!}{{{}\mathbf{h}}_{\bullet}\,\mathbf{h}!\,A(\mathbf{h})}\prod_{\mathbf{a}\in\mathsf{supp}C}\frac{\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{C(\mathbf{a})}}{{{}\mathbf{a}}_{\bullet}^{C(\mathbf{a})}C(\mathbf{a})!}$
$\displaystyle=$ $\displaystyle\
\frac{M_{\mathbf{n}}(C)}{A(\mathbf{h})}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{\mathbf{h}!}\frac{\mathbf{n}!}{(\mathbf{n}-\mathbf{h})!}$
(3.14) and $\displaystyle J(\mathbf{S};\boldsymbol{\alpha};A)=$
$\displaystyle\
J(\mathbf{S};\boldsymbol{\alpha};C)\,\big{(}{\mathbf{s}_{1}^{\diamond
h_{1}}\diamond\cdots\diamond\mathbf{s}_{q}^{\diamond
h_{q}}}\big{)}\cdot\boldsymbol{\alpha}\,\mathrm{.}$ (3.15)

Substituting (3.14) and (3.15) in (3.13) above, and simplifying $\mathbf{n}!$,

$\displaystyle\partial_{p}^{b}\,\tilde{\zeta}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=$
$\displaystyle\
\alpha_{p}\sum_{\mathbf{e}_{b}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\sum_{C\vDash\mathbf{n}-\mathbf{h}}\frac{M_{\mathbf{n}}(C)}{(\mathbf{n}-\mathbf{h})!}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{b})!}\
\mathbf{S}_{p}^{\mathbf{h}-\mathbf{e}_{b}}\,J(\mathbf{S};\boldsymbol{\alpha};C)$
$\displaystyle=$ $\displaystyle\
\alpha_{p}\sum_{\mathbf{e}_{b}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{b})!}\
\mathbf{S}_{p}^{\mathbf{h}-\mathbf{e}_{b}}\,\tilde{\zeta}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]\,\mathrm{.}$

Combining the inductive hypothesis with (3.7) and (3.9) with $a=\ell$ and
$b=j$,

$\displaystyle\partial_{p}^{b}\,\tilde{\zeta}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]=$
$\displaystyle\
\alpha_{p}\sum_{\mathbf{e}_{b}\leq_{\diamond}\mathbf{h}\leq_{\diamond}\mathbf{n}}\frac{({{}\mathbf{h}}_{\bullet}-1)!}{(\mathbf{h}-\mathbf{e}_{b})!}\
\mathbf{S}_{p}^{\mathbf{h}-\mathbf{e}_{b}}\,\tilde{\nu}_{\mathbf{n}-\mathbf{h}}[\mathbf{S};\boldsymbol{\alpha}]=\partial_{p}^{b}\,\tilde{\nu}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]\,\mathrm{.}$

By arbitrariness of $p$ and $b$ we conclude that
$\tilde{\zeta}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]-\tilde{\nu}_{\mathbf{n}}[\mathbf{S};\boldsymbol{\alpha}]$
is constant as a function of $\mathbf{S}$, hence vanishing by choosing
$\mathbf{S}=\operatorname{{\mathbf{0}}}$. ∎

##  4 A Polychromatic ESF

Let $r$ be the number of cycles of a random permutation
$\pi\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$. Assume that $\pi$ is chosen
with a probability proportional to $\theta^{r}$ for some $\theta>0$. Then, the
probability that $\pi$ has cycle structure $\boldsymbol{\lambda}\vdash n$ is
precisely the Ewens distribution $E_{\theta}(\boldsymbol{\lambda})$. We
provide a generalization of this statement to the case of colored
permutations, with coloring and cycle structure indexed by a $q$-colored
partition.

Let

$\displaystyle\mathcal{A}_{n}\operatorname{\coloneqq}\bigcup_{\mathbf{n}\in{\mathbb{N}}^{q}_{*}:{{}\mathbf{n}}_{\bullet}=n}\mathcal{A}_{\mathbf{n}}$
(4.1)

be the family of all multisets $A$ on ${\mathbb{N}}^{q}_{*}$ with
$\mathsf{shape}(A)\vdash n$.

######  Definition 4.1 (Polychromatic ESF).

Fix $n,q\in{\mathbb{N}}_{1}$, $\theta>0$, and $\mathbf{p}\in\Delta^{q-1}$. The
_polychromatic ESF_ $E_{\theta,\mathbf{p}}^{n}$ is the probability
distribution on $\mathcal{A}_{n}$ given by

$\begin{aligned}
E_{\theta,\mathbf{p}}^{n}(A)\operatorname{\coloneqq}\frac{n!}{\left\langle\theta\right\rangle_{n}}\,\theta^{{\mathsf{card}(A)}}\,\frac{\mathbf{p}^{\mathsf{col}(A)}}{\mathsf{col}(A)!}M_{\mathsf{col}(A)}(A)\end{aligned}\,\mathrm{,}\;\,\qquad
A\in\mathcal{A}_{n}\,\mathrm{.}$ (4.2)

###### Proof.

Let us verify that $E_{\theta,\mathbf{p}}^{n}$ is indeed a probability
distribution on $\mathcal{A}_{n}$. For fixed $k>n$ set
$\mathbf{s}_{j}\operatorname{\coloneqq}p_{j}{\mathbf{1}}^{\scriptscriptstyle{(k)}}$,
$j\in[q]$, and
$\boldsymbol{\alpha}\operatorname{\coloneqq}(\theta/k){\mathbf{1}}^{\scriptscriptstyle{(k)}}$.
Respectively by: the Multinomial Theorem, Theorem 3.1, and the definition
(2.7) of $Z_{\mathbf{n}}$,

$\displaystyle 1=$
$\displaystyle\sum_{\mathbf{n}\in{\mathbb{N}}_{0}^{q}:{{}\mathbf{n}}_{\bullet}=n}\binom{n}{\mathbf{n}}\mathbf{p}^{\mathbf{n}}=\sum_{\mathbf{n}\in{\mathbb{N}}_{0}^{q}:{{}\mathbf{n}}_{\bullet}=n}\binom{n}{\mathbf{n}}\int_{\Delta^{k-1}}\prod_{j}^{q}(\mathbf{s}_{j}\cdot\mathbf{x})^{n_{j}}\mathop{}\\!\mathrm{d}D_{\boldsymbol{\alpha}}(\mathbf{x})$
$\displaystyle=$
$\displaystyle\sum_{\mathbf{n}\in{\mathbb{N}}_{0}^{q}:{{}\mathbf{n}}_{\bullet}=n}\binom{n}{\mathbf{n}}\frac{\mathbf{n}!}{\left\langle\theta\right\rangle_{n}}\frac{1}{\mathbf{n}!}\sum_{A\vDash\mathbf{n}}M_{\mathbf{n}}(A)\prod_{\mathbf{a}\in\mathsf{supp}A}(\theta\,\mathbf{p}^{\mathbf{a}})^{A(\mathbf{a})}$
$\displaystyle=$
$\displaystyle\sum_{A\in\mathcal{A}_{n}}\frac{\theta^{{\mathsf{card}(A)}}}{\left\langle\theta\right\rangle_{n}}\binom{n}{\mathsf{col}(A)}\mathbf{p}^{\mathsf{col}(A)}M_{\mathsf{col}(A)}(A)\,\mathrm{.}\qed$

######  Remark 4.2 ($q=1$).

When $q=1$, we have $\mathbf{p}=p=1$ and $\mathsf{col}(A)=n$ for every
$A\in\mathcal{A}_{n}$, thus (4.2) reduces to the standard ESF by Remark 2.4.

######  Lemma 4.3 (Conditioning).

Fix $\mathbf{n}\in{\mathbb{N}}_{0}^{q}$ with ${{}\mathbf{n}}_{\bullet}=n$.
Then, the conditional probability
$E_{\theta,\mathbf{p}}^{n}\left[{\,\cdot\,}|\mathsf{col}({\,\cdot\,})=\mathbf{n}\right]$
satisfies

$E_{\theta,\mathbf{p}}^{n}\left[{\,\cdot\,}|\mathsf{col}({\,\cdot\,})=\mathbf{n}\right]=\frac{\theta^{{\mathsf{card}(A)}}}{\left\langle\theta\right\rangle_{n}}M_{\mathbf{n}}(A)\,\mathrm{,}\;\,\qquad
A\in\mathcal{A}_{\mathbf{n}}\,\mathrm{.}$ (4.3)

###### Proof.

For fixed $k>n$ set
$\mathbf{s}_{1}=\cdots=\mathbf{s}_{q}\operatorname{\coloneqq}{\mathbf{1}}^{\scriptscriptstyle{(k)}}$,
and
$\boldsymbol{\alpha}\operatorname{\coloneqq}(\theta/k){\mathbf{1}}^{\scriptscriptstyle{(k)}}$.
By Theorem 3.1 and by the definition (2.7) of $Z_{\mathbf{n}}$,

$\displaystyle
1=\int_{\Delta^{k-1}}\prod_{j}^{q}(\mathbf{s}_{j}\cdot\mathbf{x})^{n_{j}}\mathop{}\\!\mathrm{d}D_{\boldsymbol{\alpha}}=\frac{\mathbf{n}!}{\left\langle\theta\right\rangle_{n}}\frac{1}{\mathbf{n}!}\sum_{A\vDash\mathbf{n}}M_{\mathbf{n}}(A)\prod_{\mathbf{a}\in\mathsf{supp}A}\theta^{A(\mathbf{a})}\,\mathrm{,}\;\,$

hence

$\sum_{A\vDash\mathbf{n}}\theta^{{\mathsf{card}(A)}}M_{\mathbf{n}}(A)=\left\langle\theta\right\rangle_{n}\,\mathrm{.}$
(4.4)

Now,

$E_{\theta,\mathbf{p}}^{n}\left[A|\mathsf{col}(A)=\mathbf{n}\right]=\frac{E_{\theta,\mathbf{p}}^{n}(A)}{E_{\theta,\mathbf{p}}^{\mathbf{n}}\left[\mathsf{col}({\,\cdot\,})=\mathbf{n}\right]}\quad\text{if}\quad\mathsf{col}(A)=\mathbf{n}$
(4.5)

and $0$ otherwise. Furthermore,

$\displaystyle
E_{\theta,\mathbf{p}}^{n}\left[\mathsf{col}({\,\cdot\,})=\mathbf{n}\right]=$
$\displaystyle\sum_{A\vDash\mathbf{n}}E_{\theta,\mathbf{p}}^{\mathbf{n}}(A)=\sum_{A\vDash\mathbf{n}}\frac{n!}{\left\langle\theta\right\rangle_{n}}\,\theta^{{\mathsf{card}(A)}}\,\frac{\mathbf{p}^{\mathsf{col}(A)}}{\mathsf{col}(A)!}M_{\mathsf{col}(A)}(A)$
(4.6) $\displaystyle=$
$\displaystyle\frac{n!}{\left\langle\theta\right\rangle_{n}}\frac{\mathbf{p}^{\mathbf{n}}}{\mathbf{n}!}\sum_{A\vDash\mathbf{n}}\theta^{{\mathsf{card}(A)}}M_{\mathsf{col}(A)}(A)=n!\frac{\mathbf{p}^{\mathbf{n}}}{\mathbf{n}!}$

by (4.4). Combining (4.5), (4.6), and (4.2) thus yields

$E_{\theta,\mathbf{p}}^{n}\left[A|\mathsf{col}(A)=\mathbf{n}\right]=\frac{\theta^{{\mathsf{card}(A)}}}{\left\langle\theta\right\rangle_{n}}M_{\mathbf{n}}(A)\quad\text{if}\quad\mathsf{col}(A)=\mathbf{n}$

and $0$ otherwise. ∎

Since
$E_{\theta,\mathbf{p}}^{n}\left[{\,\cdot\,}|\mathsf{col}({\,\cdot\,})=\mathbf{n}\right]$
does not depend on $\mathbf{p}$, let us set

$E_{\theta}^{\mathbf{n}}\operatorname{\coloneqq}E_{\theta,\mathbf{p}}^{n}\left[{\,\cdot\,}|\mathsf{col}({\,\cdot\,})=\mathbf{n}\right]\quad\text{on}\quad\mathcal{A}_{\mathbf{n}}\,\mathrm{.}$

In analogy with the standard ESF, the conditional probability
$E_{\theta}^{\mathbf{n}}$ counts $\theta$-biased $q$-colored permutations, as
we now show.

######  Proposition 4.4.

Fix $\theta>0$ and let $\pi\in\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$ be a
$\theta$-biased random permutation. Then,

$\mathbf{P}\big{[}\Pi(\pi)=A\big{]}=E_{\theta}^{\mathbf{n}}(A)\,\mathrm{,}\;\,\qquad
A\in\mathcal{A}_{\mathbf{n}}\,\mathrm{.}$ (4.7)

###### Proof.

Let $r$ be the number of cycles of $\pi$ including fixed points. Since $\pi$
is $\theta$-biased and applying Proposition 2.13, we have

$\displaystyle\mathbf{P}\big{[}\Pi(\pi)=A\big{]}=C_{\theta}\,\theta^{r}\left\lvert\Pi^{-1}(A)\right\rvert=C_{\theta}\,\theta^{r}M_{\mathsf{col}(A)}(A)\,\mathrm{.}$

The conclusion follows since $E_{\theta}^{\mathbf{n}}$ is a probability
measure by Lemma 4.3. ∎

######  Remark 4.5.

We can rephrase Proposition 4.4 by saying that $E_{\theta}^{\mathbf{n}}$ is
the push-forward via $\Pi$ of the law $\mathbf{P}$ of a $\theta$-biased random
permutation in $\mathfrak{S}_{{{}\mathbf{n}}_{\bullet}}$. Furthermore, as a
consequence of Lemma 4.3 and Corollary 3.3, we see that

$E_{\theta}(\boldsymbol{\lambda})=\sum_{\begin{subarray}{c}A\vDash\mathbf{n}\\\
\mathsf{shape}{A}=\boldsymbol{\lambda}\end{subarray}}E_{\theta}^{\mathbf{n}}(A)\,\mathrm{,}\;\,\qquad\boldsymbol{\lambda}\vdash
n\,\mathrm{.}$ (4.8)

That is, $E_{\theta}$ is the push-forward of $E_{\theta}^{\mathbf{n}}$ via the
function $\mathsf{shape}$. In this sense, the newly defined measure
$E_{\theta}^{\mathbf{n}}$ can be seen as ‘intermediate’ between $\mathbf{P}$
and $E_{\theta}$.

Finally, let us collect here the main properties of
$E_{\theta,\mathbf{p}}^{n}$ with respect to manipulations of $\mathbf{p}$. For
each set partition
$\mathbf{L}\operatorname{\coloneqq}\left\\{L_{1},\dotsc,L_{r}\right\\}\vdash[q]$
denote by $s_{\mathbf{L}}\colon[q]\to[r]$ the $\mathbf{L}$-_degeneracy map_
defined by $s_{\mathbf{L}}^{-1}(k)=L_{k}$ for $k\in[r]$. Further let
$\mathbf{S}_{\mathbf{L}}\in\left\\{0,1\right\\}^{r\times q}$ be the matrix
$[\mathbf{S}_{\mathbf{L}}]_{i}^{j}\operatorname{\coloneqq}\operatorname{\mathbf{1}}_{j\in
s_{\mathbf{L}}^{-1}(i)}$ and note that
$\mathbf{S}_{\mathbf{L}}\colon{\mathbb{N}}^{q}_{*}\to{\mathbb{N}}^{r}_{*}$ and
$\mathbf{S}_{\mathbf{L}}\colon\Delta^{q-1}\to\Delta^{r-1}$.

Arguing similarly as in the proof of Definition 4.1, choosing
$\mathbf{s}_{j}=\mathbf{s}_{j^{\prime}}$ in (3.2) whenever $j,j^{\prime}\in
L_{i}$ for some $i$, we have the following.

######  Proposition 4.6 (Aggregation).

Let $n,q\in{\mathbb{N}}_{1}$, $\theta>0$, and $\mathbf{p}\in\Delta^{q-1}$.
Then, cf. (2.1),

${(\mathbf{S}_{\mathbf{L}})_{*}}_{\sharp}E_{\theta,\mathbf{p}}^{n}=E_{\theta,\mathbf{S}_{\mathbf{L}}\mathbf{p}}^{n}\,\mathrm{,}\;\,\qquad\mathbf{L}\vdash[q]\,\mathrm{.}$

###  4.1 A Hoppe-type urn model

In [13], F. M. Hoppe showed that the ESF $E_{\theta}$ is the marginal
distribution of a discrete-time Markov process $\left(\Pi_{t}\right)_{t}$ of
integer partitions $\Pi_{t}\vdash t$ obtained from the sampling process
$\left(X_{t}\right)_{t}$ of what is now known as _Hoppe’s urn model_. We adapt
his construction to a similar urn model, resulting in a Markov process with
values in the space of colored integer partitions and with marginal
distribution $E_{\theta,\mathbf{p}}^{t}$ at time $t$.

Denote by $\mathrm{Cat}_{\mathbf{p}}$ the categorical distribution on $[q]$
with parameters $\mathbf{p}\in\Delta^{q-1}$.

Consider a process $Y_{\circ}\operatorname{\coloneqq}\left(Y_{t}\right)_{t}$
generated by sampling from an urn containing one cube and various numbers of
labelled colored balls. At time $0$, the urn contains only the cube. At every
(integer) time $t$, the labels are consecutive and ranging in
${\mathbb{N}}_{1}$, while the colors range in $[q]$. The cube has mass
$\theta$ and every ball has mass $1$. At time $t$, an object in the urn is
selected at random with a probability proportional to its mass. If it is a
ball, it is returned together with one additional ball of the same label and
of a color chosen according to $\mathrm{Cat}_{\mathbf{p}}$ independently of
the label. If it is the cube, it is returned together with a ball with the
smallest label previously not present in the urn and of a color chosen
according to $\mathrm{Cat}_{\mathbf{p}}$. We define random variables
$r_{t}\in{\mathbb{N}}_{1}$ and $Y_{t}\in{\mathbb{N}}_{1}\times[q]$ as the
number of distinct labels (i.e. the maximal label) present in the urn, and the
label and color of the additional ball returned after the $t^{\text{th}}$
drawing. Observe that, for every $T\in{\mathbb{N}}_{1}$, the process
$Y_{\circ}$ defines a random $q$-colored partition $\mathscr{A}_{T}$ by
letting

$\displaystyle\mathbf{a}_{T}(i)\operatorname{\coloneqq}\left(a_{T,1}(i),\dotsc,a_{T,q}(i)\right)\,\mathrm{,}\;\,a_{T,j}(i)\operatorname{\coloneqq}\left\lvert\left\\{t\in[T]:Y_{t}=(i,j)\right\\}\right\rvert\,\mathrm{,}\;\,\qquad\mathscr{A}_{T}\operatorname{\coloneqq}\sum_{i}^{r_{T}}\operatorname{\mathbf{1}}_{\mathbf{a}_{T}(i)}\,\mathrm{.}$
(4.9)

As a consequence, in the notation of [13], the first component $Y_{t,1}$ of
$Y_{t}$ satisfies $Y_{t,1}=X_{t}$, while $\mathsf{shape}(\mathscr{A}_{T})$
coincides with $\Pi_{T}$. We call the Markov process $Y_{\circ}$ the
_polychromatic Hoppe urn_ (PHU), and the process
$\mathscr{A}_{\circ}\operatorname{\coloneqq}\left(\mathscr{A}_{T}\right)_{T}$
the _PHU-partition process_.

######  Proposition 4.7.

$\mathscr{A}_{\circ}$ is a Markov process with marginal distribution

$\mathbf{P}[\mathscr{A}_{T}=A]=E_{\theta,\mathbf{p}}^{T}(A)\,\mathrm{,}\;\,\qquad
A\in\mathcal{A}_{T}\,\mathrm{.}$ (4.10)

###### Proof.

The Markov property is trivially satisfied. With the notation of (4.9), the
random variables $\left({{}\mathbf{a}_{T}(i)}_{\bullet}\right)_{i}$ are
$\left(Y_{t,1}\right)_{t\leq T}$-measurable. In order to compute the marginal
distribution at time $T$, fix $A\in\mathcal{A}_{T}$, and set
$\boldsymbol{\lambda}\operatorname{\coloneqq}\mathsf{shape}(A)$ and
$r\operatorname{\coloneqq}{{}\boldsymbol{\lambda}}_{\bullet}$.

We introduce two families of functions:

$\displaystyle\mathcal{F}\coloneqq\big{\\{}\mathbf{f}:[r]\to\mathsf{supp}(A)\,\colon\,\left\lvert\mathbf{f}^{-1}(\mathbf{a})\right\rvert=A(\mathbf{a})\,\mathrm{,}\;\,\quad\mathbf{a}\in\mathsf{supp}(A)\big{\\}}\,\mathrm{,}\;\,$
$\displaystyle\mathcal{G}\coloneqq\big{\\{}g={{}({\,\cdot\,})}_{\bullet}\circ\mathbf{f}={{}\mathbf{f}({\,\cdot\,})}_{\bullet}\,\colon\,\mathbf{f}\in\mathcal{F}\big{\\}}\,\mathrm{.}$

Since the colors $Y_{t,2}$ are chosen independently of one another and of the
labels $Y_{t,1}$,

$\displaystyle\mathbf{P}$
$\displaystyle\big{[}\mathscr{A}_{T}=A\big{|}\left(Y_{t,1}\right)_{t\leq
T}\big{]}=$
$\displaystyle=\sum_{\mathbf{f}\in\mathcal{F}}\mathbf{P}\big{[}\mathbf{f}({\,\cdot\,})=\mathbf{a}_{T}({\,\cdot\,})\big{|}\left(Y_{t,1}\right)_{t\leq
T}\big{]}=\sum_{\mathbf{f}\in\mathcal{F}}\prod_{i=1}^{r}\mathbf{P}\big{[}\mathbf{f}(i)=\mathbf{a}_{T}(i)\big{|}\left(Y_{t,1}\right)_{t\leq
T}\big{]}$
$\displaystyle=\sum_{\mathbf{f}\in\mathcal{F}}\prod_{i=1}^{r}\binom{{{}\mathbf{f}(i)}_{\bullet}}{\mathbf{f}(i)}\mathbf{p}^{\mathbf{f}(i)}\operatorname{\mathbf{1}}_{\left\\{{{}\mathbf{f}(i)}_{\bullet}={{}\mathbf{a}_{T}(i)}_{\bullet}\right\\}}=\sum_{\mathbf{f}\in\mathcal{F}}\mathbf{p}^{\mathsf{col}(A)}\operatorname{\mathbf{1}}_{\left\\{{{}\mathbf{f}({\,\cdot\,})}_{\bullet}={{}\mathbf{a}_{T}({\,\cdot\,})}_{\bullet}\right\\}}\prod_{\mathbf{a}\in\mathsf{supp}(A)}\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}$
$\displaystyle=\left\lvert\left\\{\mathbf{f}\in\mathcal{F}\,\colon\,{{}\mathbf{f}({\,\cdot\,})}_{\bullet}={{}\mathbf{a}_{T}({\,\cdot\,})}_{\bullet}\right\\}\right\rvert\mathbf{p}^{\mathsf{col}(A)}\prod_{\mathbf{a}\in\mathsf{supp}(A)}\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}\,\mathrm{.}$

It can be easily checked that for every $g\in\mathcal{G}$ the following
identities hold:

$\displaystyle\left\lvert\left\\{\mathbf{f}\in\mathcal{F}:g={{}({\,\cdot\,})}_{\bullet}\circ\mathbf{f}\right\\}\right\rvert=\prod_{i}\binom{\lambda_{i}}{\left(A(\mathbf{a})\right)_{\mathbf{a}\in\mathsf{supp}(A)\,:\,{{}\mathbf{a}}_{\bullet}=i}}=\frac{\boldsymbol{\lambda}!}{\displaystyle\prod_{\mathbf{a}\in\mathsf{supp}(A)}A(\mathbf{a})!}\,\mathrm{.}$

Thus,

$\displaystyle\mathbf{P}\big{[}\mathscr{A}_{T}=A\big{|}\left(Y_{t,1}\right)_{t\leq
T}\big{]}$
$\displaystyle=\left\lvert\left\\{g\in\mathcal{G}:g({\,\cdot\,})={{}\mathbf{a}_{T}({\,\cdot\,})}_{\bullet}\right\\}\right\rvert\boldsymbol{\lambda}!\,\mathbf{p}^{\mathsf{col}(A)}\prod_{\mathbf{a}\in\mathsf{supp}(A)}\frac{\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}}{A(\mathbf{a})!}$
$\displaystyle=\operatorname{\mathbf{1}}_{\left\\{\mathsf{shape}(\mathscr{A}_{T})=\boldsymbol{\lambda}\right\\}}\boldsymbol{\lambda}!\,\mathbf{p}^{\mathsf{col}(A)}\prod_{\mathbf{a}\in\mathsf{supp}(A)}\frac{\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}}{A(\mathbf{a})!}\,\mathrm{.}$
(4.11)

Taking the expectation over $\left(Y_{t}\right)_{t\leq T}$ on both sides of
(4.11), we infer that

$\mathbf{P}[\mathscr{A}_{T}=A]=\mathbf{P}[\mathsf{shape}(\mathscr{A}_{T})=\boldsymbol{\lambda}]\,\boldsymbol{\lambda}!\,\mathbf{p}^{\mathsf{col}(A)}\prod_{\mathbf{a}\in\mathsf{supp}(A)}\frac{\binom{{{}\mathbf{a}}_{\bullet}}{\mathbf{a}}^{A(\mathbf{a})}}{A(\mathbf{a})!}\,\mathrm{.}$
(4.12)

By the formula for the marginal distribution of Hoppe’s urn model, [13, Eqn.
(1)],

$\mathbf{P}[\mathsf{shape}(\mathscr{A}_{T})=\boldsymbol{\lambda}]=\frac{T!}{\left\langle\theta\right\rangle_{T}}\prod_{i=1}^{T}\frac{\theta^{\lambda_{i}}}{i^{\lambda_{i}}\lambda_{i}!}=\frac{T!}{\boldsymbol{\lambda}!}\frac{\theta^{r}}{\left\langle\theta\right\rangle_{T}}\prod_{\mathbf{a}\in\mathsf{supp}A}\frac{1}{{{}\mathbf{a}}_{\bullet}^{A(\mathbf{a})}}\,\mathrm{.}$
(4.13)

Combining (4.12) and (4.13), the identity (4.10) follows. ∎

###  4.2 Consistency

In [16, 17], J.F.C. Kingman introduced a celebrated notion of _consistency_
for stochastic processes on partitions, and showed that a sequence of random
partitions $\left(\boldsymbol{\lambda}_{n}\right)_{n}$ with
$\boldsymbol{\lambda}_{n}\vdash n$ distributed according to $E_{\theta}$,
satisfies this notion. Precisely, if $n$ objects are partitioned into classes
with sizes given by $\boldsymbol{\lambda}_{n}$, and one object is deleted
uniformly at random, independently of $\boldsymbol{\lambda}_{n}$, the
partition of the $n-1$ remaining objects has class sizes distributed as
$\boldsymbol{\lambda}_{n-1}$, cf. e.g. [23, p. 146].

In this section, we show that the polychromatic ESF satisfies a similar
consistency property. Denote by
$\mathcal{A}\operatorname{\coloneqq}\bigcup_{n}\mathcal{A}_{n}$ the family of
all finite multisets on ${\mathbb{N}}^{q}_{*}$, and set

$A_{\setminus\mathbf{a},j}\operatorname{\coloneqq}\begin{cases}A-\operatorname{\mathbf{1}}_{\mathbf{a}}&\text{if
}\mathbf{a}=\mathbf{e}_{j}\,\mathrm{,}\\\
A-\operatorname{\mathbf{1}}_{\mathbf{a}}+\operatorname{\mathbf{1}}_{\mathbf{a}-\mathbf{e}_{j}}&\text{otherwise}\end{cases}\,\mathrm{,}\;\,\qquad\mathbf{a}\in\mathsf{supp}A\,\mathrm{,}\;\,j\in[q]\,\mathrm{.}$

Following [17], we define a system $S=S_{nm}$, $n\in{\mathbb{N}}_{1}$, $m\leq
n$, of probability kernels on $\mathcal{A}$. Firstly, set

$\displaystyle S(A,B)\operatorname{\coloneqq}$ $\displaystyle\
\operatorname{\mathbf{1}}_{A=B}\,\mathrm{,}\;\,$ $\displaystyle A,B$
$\displaystyle\in\mathcal{A}_{n}\,\mathrm{,}\;\,$ (4.14a) $\displaystyle
S(A,B)\operatorname{\coloneqq}$ $\displaystyle\
\begin{cases}\frac{a_{j}A(\mathbf{a})}{n}&\text{if
}B=A_{\setminus\mathbf{a},j}\,\mathrm{,}\\\
0&\text{otherwise}\end{cases}\,\mathrm{,}\;\,$ $\displaystyle
A\in\mathcal{A}_{n}\,\mathrm{,}\;\,$ $\displaystyle
B\in\mathcal{A}_{n-1}\,\mathrm{,}\;\,$ (4.14b)

and note that $S(A,{\,\cdot\,})$ is a probability on $\mathcal{A}_{n-1}$ for
every $A\in\mathcal{A}_{n}$. Secondly, let $S$ be the unique system of kernels
extending (4.14) and satisfying the cocycle relation

$S(A,C)=\sum_{B\in\mathcal{A}_{m}}S(A,B)\,S(B,C)\,\mathrm{,}\;\,\qquad
A\in\mathcal{A}_{n}\,\mathrm{,}\;\,C\in\mathcal{A}_{\ell}\,\mathrm{,}\;\,\quad\ell<m<n\,\mathrm{.}$
(4.15)

Note that $S_{nm}(A,{\,\cdot\,})$ is a probability on $\mathcal{A}_{m}$ for
every $m$ and every $A\in\mathcal{A}_{n}$, since it is so for $m=n-1$ as noted
above, and in light of (4.15).

######  Remark 4.8.

Analogously to the case of usual integer partitions, the system $S$ may be
interpreted as the selection of a random sampling (uniform, without
replacement) of $m$ elements from a given $q$-colored partition
$A\in\mathcal{A}_{n}$, resulting in the $q$-colored partition
$B\in\mathcal{A}_{m}$. The cocycle relation (4.15) is then a consequence of
the consistency of random sub-sampling.

Let us now turn to probability measures on $\mathcal{A}$. For
$n\in{\mathbb{N}}_{1}$ let $\mathscr{P}(\mathcal{A}_{n})$ be the set of all
probability measures on $\mathcal{A}_{n}$. Define a system $\sigma$ of maps
$\sigma_{nm}\colon\mathscr{P}(\mathcal{A}_{n})\to\mathscr{P}(\mathcal{A}_{m})$
by

$\big{(}{\sigma_{nm}\mathbf{P}}\big{)}(B)\longmapsto\mathbf{P}[S({\,\cdot\,},B)]\,\mathrm{,}\;\,$

and note that $\sigma$ satisfies the cocycle relation

$\sigma_{n\ell}=\sigma_{m\ell}\circ\sigma_{nm}\,\mathrm{,}\;\,\qquad\ell<m<n\,\mathrm{.}$
(4.16)

######  Definition 4.9 (Consistency).

We say that a family $\left(\mathbf{P}_{n}\right)_{n}$ of probability measures
$\mathbf{P}_{n}$ on $\mathcal{A}_{n}$ is _consistent_ (w.r.t. the system
$\sigma$) if $\mathbf{P}_{m}=\sigma_{nm}\mathbf{P}_{n}$ for every $m\leq n$.

######  Theorem 4.10.

For every $\theta>0$ and $\mathbf{p}\in\Delta^{q-1}$ the family
${\big{(}E_{\theta,\mathbf{p}}^{n}\big{)}}_{n}$ is consistent.

###### Proof.

In light of (4.16), it suffices to verify that
$\sigma_{nm}E_{\theta,\mathbf{p}}^{n}=E_{\theta,\mathbf{p}}^{m}$ for $m=n-1$
and for every $n$. To this end, let $\mathbf{Q}$ be the law of the PHU
partition $\mathscr{A}_{\circ}$ on its path space. By Bayes formula, and
Proposition 4.7,

$\displaystyle\mathbf{Q}[\mathscr{A}_{n-1}=B\mid\mathscr{A}_{n}=A]=$
$\displaystyle\
\frac{\mathbf{Q}[\mathscr{A}_{n}=A\mid\mathscr{A}_{n-1}=B]\,\mathbf{Q}[\mathscr{A}_{n-1}=B]}{\mathbf{Q}[\mathscr{A}_{n}=A]}$
$\displaystyle=$ $\displaystyle\
\frac{\mathbf{Q}[\mathscr{A}_{n}=A\mid\mathscr{A}_{n-1}=B]\,E_{\theta,\mathbf{p}}^{n-1}(B)}{E_{\theta,\mathbf{p}}^{n}(A)}\,\mathrm{.}$
(4.17)

Furthermore, it follows from the definition of $\mathscr{A}_{\bullet}$ that

$\displaystyle\mathbf{Q}[\mathscr{A}_{n}=A\mid\mathscr{A}_{n-1}=B]=$
$\displaystyle\sum_{\mathbf{a}\in\mathsf{supp}A}\sum_{\begin{subarray}{c}j\in[q]:\\\
\mathbf{e}_{j}\leq_{\diamond}\mathbf{a},\mathbf{e}_{j}\neq\mathbf{a}\end{subarray}}\operatorname{\mathbf{1}}_{A=B+\operatorname{\mathbf{1}}_{\mathbf{a}}-\operatorname{\mathbf{1}}_{\mathbf{a}-\mathbf{e}_{j}}}\frac{{{}\mathbf{a}}_{\bullet}-1}{\theta+n-1}B(\mathbf{a}-\mathbf{e}_{j})\,p_{j}$
(4.18)
$\displaystyle+\sum_{j=1}^{q}\operatorname{\mathbf{1}}_{A=B+\operatorname{\mathbf{1}}_{\mathbf{e}_{j}}}\frac{\theta}{\theta+n-1}\,p_{j}\,\mathrm{.}$

On the other hand, by definition (4.2) of $E_{\theta,\mathbf{p}}^{n}$,

$\displaystyle\frac{E_{\theta,\mathbf{p}}^{n-1}(B)}{E_{\theta,\mathbf{p}}^{n}(A)}=\begin{cases}\displaystyle\frac{\theta+n-1}{np_{j}}\frac{a_{j}}{{{}\mathbf{a}}_{\bullet}-1}\frac{A(\mathbf{a})}{A(\mathbf{a}-\mathbf{e}_{j})+1}&\text{if
}A=B+\operatorname{\mathbf{1}}_{\mathbf{a}}-\operatorname{\mathbf{1}}_{\mathbf{e}_{j}}\\\
\displaystyle\frac{\theta+n-1}{\theta np_{j}}A(\mathbf{e}_{j})&\text{if
}A=B+\operatorname{\mathbf{1}}_{\mathbf{e}_{j}}\end{cases}$ (4.19)

Combining (4.17)–(4.19), we thus have

$\displaystyle\mathbf{Q}[\mathscr{A}_{n-1}=B\mid\mathscr{A}_{n}=A]=$
$\displaystyle\sum_{\mathbf{a}\in\mathsf{supp}A}\sum_{j\in[q]:\mathbf{e}_{j}<\mathbf{a}}\operatorname{\mathbf{1}}_{A=B+\operatorname{\mathbf{1}}_{\mathbf{a}}-\operatorname{\mathbf{1}}_{\mathbf{a}-\mathbf{e}_{j}}}\frac{a_{j}A(\mathbf{a})}{n}+\sum_{j=1}^{q}\operatorname{\mathbf{1}}_{A=B+\operatorname{\mathbf{1}}_{\mathbf{e}_{j}}}\frac{A(\mathbf{e}_{j})}{n}$
$\displaystyle=$
$\displaystyle\sum_{\mathbf{a}\in\mathsf{supp}A}\sum_{j}^{q}\operatorname{\mathbf{1}}_{A_{\setminus\mathbf{a},j}=B}\frac{a_{j}A(\mathbf{a})}{n}=S_{n\
n-1}(A,B)\,\mathrm{.}$

Finally, respectively by: the definition of $\sigma$, the previous equality
and Proposition 4.7, the law of total probability, and again Proposition 4.7,

$\displaystyle\big{(}{\sigma_{n\ n-1}E_{\theta,\mathbf{p}}^{n}}\big{)}(B)=$
$\displaystyle\sum_{A\in\mathcal{A}_{n}}S_{n\
n-1}(A,B)\,E_{\theta,\mathbf{p}}^{n}(A)=\sum_{A\in\mathcal{A}_{n}}\mathbf{Q}[\mathscr{A}_{n-1}=B\mid\mathscr{A}_{n}=A]\,\mathbf{Q}[\mathscr{A}_{n}=A]$
$\displaystyle=$ $\displaystyle\
\mathbf{Q}[\mathscr{A}_{n-1}=B]=E_{\theta,\mathbf{p}}^{n-1}(B)\,\mathrm{.}\qed$

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