Eulerian polynomials via the Weyl algebra action

Abstract

Through the action of the Weyl algebra on the geometric series, we establish a generalization of the Worpitzky identity and new recursive formulae for a family of polynomials including the classical Eulerian polynomials. We obtain an extension of the Dobiński formula for the sum of rook numbers of a Young diagram by replacing the geometric series with the exponential series. Also, by replacing the derivative operator with the q-derivative operator, we extend these results to the q-analogue setting including the q-hit numbers. Finally, a combinatorial description and a proof of the symmetry of a family of polynomials introduced by one of the authors are provided.

1 Introduction

This paper is mainly motivated by the idea of developing a theory for Eulerian polynomials and their generalizations through the formalism of the Weyl algebra. Our starting point is a family of polynomials, occasionally called hit polynomials [4, 5], already covered in Riordan’s book [16] in the late 1950s, and introduced by Kaplansky and Riordan [14]. Among other reasons, hit polynomials are interesting because of their combinatorial properties linked to rook numbers. Let us recall some notions and briefly describe the context. A non-attacking rook placement on a board D is a set P of boxes of D with no two boxes in the same row or column. The number $r_k(D)$ of non-attacking rook placements P on D with $|P|=k$ is said to be the k-th rook number of D. If $D=D_{\lambda }$ is the Young diagram of a partition $\lambda$, then we write $r_k(\lambda )$ for the k-th rook number of $D_\lambda$. In particular, for the staircase partition $\delta _{n}:=(n,n-1,\ldots ,1)$, it is well-known that the rook numbers $r_k(\delta _{n-1})$ are the Stirling numbers of the second kind $S(n,n-k)$. In this sense, the sum $R_\lambda =\sum _{k}r_k(\lambda )$ can be regarded as a generalized Bell number. By identifying the permutations in the symmetric group ${\mathfrak {S}}_n$ with the placements on the square diagram $D_n$ consisting of n rows of length n, for any partition $\lambda$ such that $D_\lambda \subseteq D_n$, we set

$$\begin{aligned}{\mathcal {A}}_{n,\lambda }(x):=\sum _{\sigma \in {\mathfrak {S}}_n}x^{|\sigma \cap D_\lambda |}.\end{aligned}$$

The polynomials ${\mathcal {A}}_{n,\lambda }(x)$ often occur within the well developed literature on rook theory [4, 6, 9,10,11,12,13,14]. It is well-known that the classical Eulerian polynomials $A_n(x)$ arise as ${\mathcal {A}}_{n,\delta _{n-1}}(x)$. In Sect. 3, we will show that ${\mathcal {A}}_{n,\delta _{n-r}}(x)$ agrees with the polynomial $^{r}\!A_n(x)$ introduced by Foata and Schützenberger [7]. This connection motivates a generalized notion of the excedance statistic that allows another combinatorial description of the polynomial ${\mathcal {A}}_{n,\lambda }(x)$. A classical formula of Frobenius, relating the Stirling numbers of the second kind and the Eulerian polynomials, extends in a straightforward manner to the following identity [4]

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x)=\sum _{k\ge 0}r_k(\lambda )\,(n-k)!\,(x-1)^k. \end{aligned}$$

(1)

Based on a q-analogue of rook numbers, Garsia and Remmel [8] provided a q-analogue for the polynomials ${\mathcal {A}}_{n,\lambda }(x)$ that generalizes identity (1). Dworkin [5] further studied the recursive properties of such polynomials and also gave a direct combinatorial interpretation of their coefficients, the q-hit numbers.

In the seventies, Navon [15] showed that rook placements also provide a natural combinatorial framework for the algebras generated by annihilation and creation operators, and in particular for the so-called normal ordering problem [2, 3, 17]. Recall that, if ${\mathbf {X}}$ denotes the operator of multiplication by x, and ${\mathbf {D}}=\frac{d}{dx}$ denotes the usual derivative operator, then ${\mathbf {D}}{\mathbf {X}}-{\mathbf {X}}{\mathbf {D}}=1$ and the algebra generated by ${\mathbf {X}}$ and ${\mathbf {D}}$ is referred to as the Weyl algebra. The normal ordering of any product ${\varvec{\Pi }}$ involving a occurrences of the operator ${\mathbf {X}}$ and b occurrences of the operator ${\mathbf {D}}$ is given by

$$\begin{aligned}{\varvec{\Pi }}=\sum _{k\ge 0}r_k(\lambda )\,{\mathbf {X}}^{a-k}{\mathbf {D}}^{b-k},\end{aligned}$$

where $\lambda$ is a suitable partition associated with ${\varvec{\Pi }}$. In this setting, the Stirling numbers of the second kind arise as the normal ordering coefficients of ${\varvec{\Pi }}=({\mathbf {X}}{\mathbf {D}})^n$.

We show that the polynomials ${\mathcal {A}}_{n,\lambda }(x)$ naturally describe the action of any product of the operators ${\mathbf {D}}$ and ${\mathbf {X}}$ on the geometric series $1/(1-x)$. More precisely, given a partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$, we define an operator ${\varvec{\Pi }}_\lambda$ such that for any square diagram $D_n$ containing $D_\lambda$,

$$\begin{aligned}{\varvec{\Pi }}_\lambda {\mathbf {D}}^{n-\lambda _1}\frac{1}{1-x}=\frac{{\mathcal {A}}_{n,\lambda ^{(n)}}(x)}{(1-x)^{n+1}},\end{aligned}$$

where $\lambda ^{(n)}$ is a partition that we call the reduced complement of $\lambda$ in $D_n$ (Theorem 5). A first consequence of this point of view is that the polynomials of Garsia and Remmel arise when the operator ${\varvec{\Pi }}_{\lambda ,q}{\mathbf {D}}_q^{n-\lambda _1}$, obtained from ${\varvec{\Pi }}_\lambda {\mathbf {D}}^{n-\lambda _1}$ by replacing ${\mathbf {D}}$ with the q-derivative ${\mathbf {D}}_q$, acts on $1/(1-x)$. More precisely, they are the polynomials ${\mathcal {A}}_{n,\lambda }(x,q)$ such that

$$\begin{aligned}{\varvec{\Pi }}_{\lambda ,q}{\mathbf {D}}_q^{n-\lambda _1}\frac{1}{1-x}=\frac{{\mathcal {A}}_{n,\lambda ^{(n)}}(x,q)}{(1-x)(1-xq)\cdots (1-xq^n)}.\end{aligned}$$

In addition, straightforward manipulations of derivatives and formal power series allow us to establish a generalization of the classical Worpitzky identity (Corollary 6), a remarkably and seemingly new property of the polynomials ${\mathcal {A}}_{n,\lambda }(x)$ with respect to derivation (Corollary 7), and a recursion formula to compute ${\mathcal {A}}_{n,\lambda }(x)$ (Corollary 8). When $\lambda =\delta _{n-r}$ a new recursive formula relating the polynomials $^r\!A_n(x)$ and the classical Eulerian polynomials is obtained. In turn, each of these results provide a corresponding q-analogue simply by replacing ${\mathbf {D}}$ with ${\mathbf {D}}_q$ (Corollaries 9,10,11). Furthermore, by letting ${\varvec{\Pi }}_\lambda {\mathbf {D}}^{n-\lambda _1}$ act on the formal power series expansion of $e^x$, we recover an extension of the classical Dobiński formula for the Bell numbers (identity (27)), and its q-analogue (identity (28)). Finally, we provide a combinatorial description and a proof of the symmetry property of the polynomials $A_{r,s,n}(x)$ (Proposition 13), defined by

$$\begin{aligned}({\mathbf {X}}^r{\mathbf {D}}^s)^n\frac{1}{1-x}=\frac{A_{r,s,n}(x)}{(1-x)^{sn+1}},\end{aligned}$$

and introduced by one of the authors of the present paper [1].

2 Partitions and rook numbers

By a partition, we mean a finite non-increasing vector $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ of positive integers called parts of $\lambda$. The number of parts of $\lambda$ is called the length of $\lambda$, and denoted by $\ell (\lambda )$. The Young diagram (or Ferrers board) of $\lambda$ is a left-aligned array of boxes, displayed in $\ell (\lambda )$ rows consisting of $\lambda _1,\lambda _2,\ldots ,\lambda _l$ boxes, from top to bottom. In analogy with matrix notation, given a Young diagram D, we let $D_{i,j}$ denote the box of D occurring at the i-th row (counting from top to bottom) and at the j-th column (counting from left to right). For instance, the Young diagram of $\lambda =(4,4,4,2,2,1)$ is shown in Fig. 1A, with a bullet drawn in the box $D_{3,2}$. The conjugate of $\lambda$ is the partition $\lambda '$ whose diagram $D_{\lambda '}$ is obtained by reflecting $D_\lambda$ with respect to its main diagonal. For example, the conjugate of $\lambda =(4,4,4,2,2,1)$ is $\lambda '=(6,5,3,3)$ and its Young diagram is shown in Fig. 1B. The border of a Young diagram D is by definition the subset of those sides lying at the rightmost position in a row, or at a lowest position in a column. The border of $D_{(4,4,4,2,2,1)}$ is highlighted in Fig. 1c.

Fig. 1

Young diagrams and their border

Given any vectors ${\varvec{r}}=(r_1,r_2,\ldots ,r_k)$ and ${\varvec{u}}=(u_1,u_2,\ldots ,u_k)$ of positive integers, we let $\lambda _{{\varvec{r}},{\varvec{u}}}$ denote the unique partition whose Young diagram has border with horizontal strips of lengths $r_1,r_2,\ldots ,r_k$ (from left to right), and vertical strips of lengths $u_1,u_2,\ldots ,u_k$ (from bottom to top). For instance, we have $\lambda _{(1,1,2),(1,2,3)}=(4,4,4,2,2,1)$ as one may check from the horizontal and vertical strips in Fig. 2.

Fig. 2

Horizontal and vertical strips of a border

Given two partitions $\lambda$ and $\mu$, we write $\lambda \subseteq \mu$ to mean that $D_\lambda \subseteq D_\mu$. Moreover, we let $D_n$ denote the square Young diagram of n rows, and for any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ such that $D_\lambda \subseteq D_n$, we call reduced complement of $\lambda$ in $D_n$ the partition $\lambda ^{(n)}:=(n-\lambda _l,n-\lambda _{l-1},\ldots ,n-\lambda _1)$. In terms of Young diagrams, $D_{\lambda ^{(n)}}$ is obtained from $D_n$ by removing the boxes of $D_\lambda$, deleting all the rows of $D_n$ lying below $D_\lambda$, then rotating by $180^\circ$. For instance, the reduced complement of (2, 2, 1) in $D_4$ is (3, 2, 2) and of (6, 6, 3, 3) in $D_9$ is (6, 6, 3, 3). They are obtained by rotating the white diagrams in Fig. 3.

Fig. 3

The reduced complement (white boxes) of a partition (dark gray boxes)

A non attacking rook placement on a Young diagram D, simply placement from now on, is a set P of blocks of D with no two boxes occurring in the same row or column. The number of placements on $D_\lambda$ consisting of k boxes, usually called the k-th rook number of $\lambda$, will be denoted by $r_k(\lambda )$. For instance, we have $r_3(4,3,1)=4$ and indeed the four placements of three boxes on $D_{(4,3,1)}$ are depicted in Fig. 4.

Fig. 4

A bullet is marked in each box of the placement

A placement of n boxes on $D_n$ can be identified with a permutation matrix of order n. Thus, denoting the symmetric group of degree n by ${\mathfrak {S}}_n$, we will consider the permutation $\sigma =\sigma _1\sigma _2\ldots \sigma _n$ and the placement $\{D_{1,\sigma (1)},D_{2,\sigma (2)},\ldots ,D_{n,\sigma (n)}\}$ on $D=D_n$ as the same object. For instance, we identify the permutations 123, 132, 213, 231, 312, 321 in ${\mathfrak {S}}_3$ with the following placements on $D_3$:

Note that $\sigma ^{-1}$ is obtained by reflecting $\sigma$ in the main diagonal of $D_n$. Hence, for all $\sigma \in {\mathfrak {S}}_n$ and for all $\lambda$ such that $D_\lambda \subseteq D_n$ we have

$$\begin{aligned} \big |\sigma \cap D_\lambda \big |=\big |\sigma ^{-1}\cap D_{\lambda '}\big |. \end{aligned}$$

(2)

Moreover, given $\sigma \in {\mathfrak {S}}_n$, let $\sigma ^\lambda =\sigma ^\lambda _1\sigma ^\lambda _2\ldots \sigma ^\lambda _n$ be defined by

$$\begin{aligned} \sigma ^\lambda _i:={\left\{ \begin{array}{ll} n+1-\sigma _{\ell (\lambda )+1-i}&{}\text { if }1\le i\le \ell (\lambda );\\ n+1-\sigma _{n+1+\ell (\lambda )-i}&{}\text { if }\ell (\lambda )+1\le i\le n. \end{array}\right. } \end{aligned}$$

(3)

It is easy to deduce that $\sigma \mapsto \sigma ^\lambda$ is a bijective map. Now, set

$$\begin{aligned}A_\lambda :=\{D_{i,j}\,|\,1\le i\le \ell (\lambda ),\,1\le j\le n\} \text { and } B_\lambda :=D\setminus A_\lambda .\end{aligned}$$

Observe that $\sigma ^\lambda$ is obtained by separately rotating by $180^\circ$ the rectangles $A_\lambda$ and $B_\lambda$ (with respect to their center). For instance, let $\lambda =(2,2,1)$, $n=5$ and $\sigma =13425$, then we have $\sigma ^\lambda =23514$ as depicted in Fig. 5.

Fig. 5

$D_\lambda$ in dark gray, $B_\lambda$ in light gray

As $\big |\sigma \cap A_\lambda \big |=\ell (\lambda )$, we obtain

$$\begin{aligned} \big |\sigma \cap D_\lambda \big |=\ell (\lambda )-\big |\sigma ^\lambda \cap D_{\lambda ^{(n)}}\big |. \end{aligned}$$

(4)

3 Generalized Eulerian polynomials

Given a partition $\lambda$, and a positive integer n such that $D_\lambda \subseteq D_n$, we define the polynomial ${\mathcal {A}}_{n,\lambda }(x)$ as follows:

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x):=\sum _{\sigma \in {\mathfrak {S}}_n}x^{|\sigma \cap D_\lambda |}. \end{aligned}$$

(5)

Moreover, we set

$$\begin{aligned} {\mathcal {A}}_{n,k,\lambda }:=\big |\{\sigma \in {\mathfrak {S}}_n\,:\,|\sigma \cap D_\lambda |=k\}\big |, \text { for }k=0,1,\ldots ,n, \end{aligned}$$

(6)

and obtain

$$\begin{aligned}{\mathcal {A}}_{n,\lambda }(x):=\sum _{k\ge 0}{\mathcal {A}}_{n,k,\lambda }\,x^k.\end{aligned}$$

Example 1

Let $\lambda =(2,2,1)$ and $n=3$. In order to obtain ${\mathcal {A}}_{3,(2,2,1)}(x)$, we compute the cardinality of $\sigma \cap D_\lambda$, for each $\sigma \in {\mathfrak {S}}_3$.

We get ${\mathcal {A}}_{3,(2,2,1)}(x)=4x^2+2x$. Note that by reflecting with respect to the main diagonal of $D_3$ (i.e., taking images under the bijection $\sigma \mapsto \sigma ^{-1}$) one obtains ${\mathcal {A}}_{3,(3,2)}(x)=4x^2+2x={\mathcal {A}}_{3,\lambda '}(x),$

Proposition 1

Given a partition $\lambda$ and a positive integer n such that $D_\lambda \subseteq D_n$, we have

• (i) ${\mathcal {A}}_{n,\lambda }(1)=n!$;

• (ii) ${\mathcal {A}}_{n,\lambda '}(x)={\mathcal {A}}_{n,\lambda }(x)$;

• (iii) ${\mathcal {A}}_{n,\lambda ^{(n)}}(x)=x^{\ell (\lambda )}{\mathcal {A}}_{n,\lambda }(1/x)$.

Proof

From (5) and (2), we have (i) and (ii), respectively. Moreover, by means of $\sigma \mapsto \sigma ^\lambda$ and (4) we have

$$\begin{aligned}x^{\ell (\lambda )}{\mathcal {A}}_{n,\lambda }(1/x)=\sum _{\sigma \in {\mathfrak {S}}_n}x^{\ell (\lambda )-|\sigma \cap D_\lambda |}=\sum _{\sigma \in {\mathfrak {S}}_n}x^{|\sigma ^\lambda \cap D_{\lambda ^{(n)}}|}={\mathcal {A}}_{n,\lambda ^{(n)}}(x),\end{aligned}$$

which gives (i). $\square$

Note that (iii) means that the coefficients of ${\mathcal {A}}_{n,\lambda }(x)$, read in decreasing order of degree, agree with the coefficients of ${\mathcal {A}}_{n,\lambda ^{(n)}}(x)$, read in increasing order of degree. For instance, if $\lambda =(3,3,2,1)$ then $\lambda ^{(7)}=(6,5,4,4)$ and in fact we have

$$\begin{aligned}{\mathcal {A}}_{7,(3,3,2,1)}(x)=192x^3+1704x^2+2496x+648\end{aligned}$$

and

$$\begin{aligned}{\mathcal {A}}_{7,(6,5,4,4)}(x)=648x^4+2496x^3+1704x^2+192x.\end{aligned}$$

In particular, the following symmetry property holds.

Corollary 2

Let n be a positive integer and $\lambda$ a partition such that $D_\lambda \subseteq D_n$. If $\lambda ^{(n)}=\lambda$ then

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x)=x^{\ell (\lambda )}{\mathcal {A}}_{n,\lambda }(1/x). \end{aligned}$$

(7)

Moreover, if $(\lambda ')^{(n)}=\lambda '$ then

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x)=x^{\lambda _1}{\mathcal {A}}_{n,\lambda }(1/x). \end{aligned}$$

(8)

Proof

Identity (7) follows from $\lambda =\lambda ^{(n)}$ and (iii). Identity (8) follows from (iii) taking into account that $\ell (\lambda ')=\lambda _1$. $\square$

An explicit expansion of ${\mathcal {A}}_{n,\lambda }(x)$ in terms of the basis $\{(x-1)^i\,|\,i\ge 0\}$ has been known since [14], where it is proved by using the inclusion–exclusion principle. Here, we provide an alternative and explicit proof.

Theorem 3

Given a partition $\lambda$ and a positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x)=\sum _{i\ge 0}r_i(\lambda )\,(n-i)!\,(x-1)^i . \end{aligned}$$

(9)

Proof

By (5) we have

$$\begin{aligned}{\mathcal {A}}_{n,\lambda }(x+1)=\sum _{\sigma \in {\mathfrak {S}}_n}(x+1)^{|\sigma \cap D_\lambda |}=\sum _{(\sigma ,B)\in \mathrm {Pairs}}x^{|B|},\end{aligned}$$

where $\mathrm {Pairs}$ denotes the set of all $(\sigma ,B)$ such that $\sigma \in {\mathfrak {S}}_n$ and $B\subseteq (\sigma \cap D_\lambda )$. Note that for all $(\sigma ,B)\in \mathrm {Pairs}$, B is a placement on $D_\lambda$. Now, for any given placement $B_0$ on $D_\lambda$, let us count the pairs $(\sigma ,B)$ such that $B=B_0$. Assume $|B_0|=i$ and consider the permutation $\sigma ^{B_0}$ obtained by adding to $B_0$ the $n-i$ available boxes on the main diagonal of $D:=D_n$, that is

$$\begin{aligned}\sigma ^{B_0}:=B_0\,\cup \,\{D_{i,i}\,|\,D_{i,j}\notin B_0 \text { for all }j=1,2,\ldots ,n\}.\end{aligned}$$

Clearly $(\sigma ^{B_0},B_0)\in \mathrm {Pairs}$. Moreover, we obtain all the pairs of type $(\sigma ,B_0)$ by permuting the $n-i$ columns of D with no boxes in $\sigma ^{B_0}\setminus B_0$. As there are $r_i(\lambda )$ placements B on $D_\lambda$ with $|B|=i$, the number of pairs $(\sigma ,B)$ such that $|B|=i$ is $r_i(\lambda )\,(n-i)!$. We recover

$$\begin{aligned}{\mathcal {A}}_{n,\lambda }(x+1)=\sum _{i\ge 0}r_i(\lambda )\,(n-i)!\,x^i,\end{aligned}$$

which gives (9) when x is replaced by $x-1$. $\square$

Example 2

Let r be a nonnegative integer. Following Foata and Schützenberger [7], we consider the polynomial

$$\begin{aligned}{}^r\!A_n(x):=\sum _{\sigma \in {\mathfrak {S}}_n}x^{\mathrm {exc}_r(\sigma )},\end{aligned}$$

where

$$\begin{aligned}\mathrm {exc}_r(\sigma ):=\big |\{i\,|\,1\le i\le n,\,\sigma _i\ge i+r\}\big |.\end{aligned}$$

Clearly, ${}^1\!A_n(x)$ is the classical Eulerian polynomial. Now, let $\sigma \mapsto \sigma '$ denote the bijection defined on ${\mathfrak {S}}_{n+r}$ by $\sigma '_i:=n+r+1-\sigma _i$, for $i=1,2,\ldots n+r$. Observe that $\sigma _i\le n+1-i$ if and only if $\sigma '_i\ge r+i$. As a consequence, we obtain

$$\begin{aligned} {\mathcal {A}}_{n+r,\delta _n}(x)=\sum _{\sigma \in {\mathfrak {S}}_{n+r}}x^{|\sigma \cap D_{\delta _n}|}=\sum _{\sigma \in {\mathfrak {S}}_{n+r}}x^{\mathrm {exc}_r(\sigma ')}={}^r\!\!A_{n+r}(x), \end{aligned}$$

(10)

or equivalently ${}^r\!\!A_{n}(x)={\mathcal {A}}_{n,\delta _{n-r}}(x)$. From (9), we recover the following Frobenius identity for the polynomials ${}^r\!\!A_{n}(x)$ [7]:

$$\begin{aligned}{}^r\!\!A_{n}(x)=\sum _{k\ge 0}\,S(n+1-r,n+1-r-k)\,(n-k)!\,(x-1)^k.\end{aligned}$$

The following generalization of the notion of excedance is motivated by Example 2. Given a partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$, a positive integer n such that $D_\lambda \subseteq D_n$, and a permutation $\sigma =\sigma _1\sigma _2\ldots \sigma _n\in {\mathfrak {S}}_n$, we set

$$\begin{aligned} \mathrm {exc}_\lambda (\sigma ):=\big |\{i\,|\,1\le i\le n,\,\sigma _i>n+1-\lambda _i\}\big |, \end{aligned}$$

(11)

where $\lambda _i=0$ is assumed for $\ell (\lambda )<i\le n$. As before, the complement bijection $\sigma \mapsto \sigma '$ provides

$$\begin{aligned}\big |\sigma \cap D_\lambda \big |=\mathrm {exc}_\lambda (\sigma '),\end{aligned}$$

so that we get

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x)=\sum _{\sigma \in {\mathfrak {S}}_n}x^{\mathrm {exc}_\lambda (\sigma )}. \end{aligned}$$

(12)

4 The Weyl algebra action

Let ${\mathbf {D}},{\mathbf {X}}:{\mathbb {Z}}[x]\rightarrow {\mathbb {Z}}[x]$ denote the derivative operator and the operator of multiplication by x, respectively. As ${\mathbf {D}}{\mathbf {X}}-{\mathbf {X}}{\mathbf {D}}=1$ the following normal ordering problem may be posed: given any product ${\varvec{\Pi }}$ involving a occurrences of the operator ${\mathbf {D}}$ and b occurrences of the operator ${\mathbf {X}}$, find the coefficients $c_i({\varvec{\Pi }})$ satisfying

$$\begin{aligned}{\varvec{\Pi }}=\sum _{i\ge 0}c_i({\varvec{\Pi }})\,{\mathbf {X}}^{b-i}{\mathbf {D}}^{a-i}.\end{aligned}$$

A beautiful answer to this problem was given by Navon [15] in terms of placements on Young diagrams. Here, we recast Navon’s result following the work of Varvak [17]. For any partition $\lambda$, we set

$$\begin{aligned} {\varvec{\Pi }}_\lambda :={\mathbf {D}}^{r_1}{\mathbf {X}}^{u_1}{\mathbf {D}}^{r_2}{\mathbf {X}}^{u_2}\cdots {\mathbf {D}}^{r_k}{\mathbf {X}}^{u_k}, \end{aligned}$$

(13)

where ${\varvec{r}}=(r_1,r_2,\ldots ,r_k)$ and ${\varvec{u}}=(u_1,u_2,\ldots ,u_k)$ are the unique vectors satisfying $\lambda =\lambda _{{\varvec{r}},{\varvec{u}}}$. Note that $\lambda _1=r_1+r_2+\cdots +r_k$ and $\ell (\lambda )=u_1+u_2+\cdots +u_k$.

Theorem 4

For any partition $\lambda$, we have

$$\begin{aligned} {\varvec{\Pi }}_\lambda =\sum _{i\ge 0}r_i(\lambda )\,{\mathbf {X}}^{\ell (\lambda )-i}{\mathbf {D}}^{\lambda _1-i}. \end{aligned}$$

(14)

Proof

Let $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$. A straightforward computation shows that ${\varvec{\Pi }}_\lambda \,1=r_{\lambda _1}(\lambda )\,x^{\ell (\lambda )-\lambda _1}$. Set

$$\begin{aligned}\mu :=(\underbrace{\lambda _1,\lambda _1,\ldots ,\lambda _1}_{m+1},\lambda _2,\ldots ,\lambda _l) \text { and }\mu \setminus \lambda :=(\underbrace{\lambda _1,\lambda _1,\ldots ,\lambda _1}_{m}).\end{aligned}$$

It follows that ${\varvec{\Pi }}_\lambda \,x^m={\varvec{\Pi }}_\lambda {\mathbf {X}}^m\,1={\varvec{\Pi }}_\mu \,1=r_{\lambda _1}(\mu )\,x^{m+\ell (\lambda )-\lambda _1}$. On the other hand, we may compute $r_{\lambda _1}(\mu )$ in the following alternative way,

$$\begin{aligned}r_{\lambda _1}(\mu )=\sum _{k\ge 0}r_k(\lambda )\,r_{\lambda _1-i}(\mu \setminus \lambda )=\sum _{i\ge 0}r_i(\lambda )\,\frac{m!}{(m-\lambda _1-i)!}.\end{aligned}$$

Then, we conclude

$$\begin{aligned}\sum _{i\ge 0}r_i(\lambda )\,{\mathbf {X}}^{\ell (\lambda )-i}{\mathbf {D}}^{\lambda _1-i}x^m=r_{\lambda _1}(\mu )\,x^{m+\ell (\lambda )-\lambda _1}={\varvec{\Pi }}_\lambda \,x^m.\end{aligned}$$

$\square$

The following theorem makes explicit the connection between the Weyl algebra and the polynomials ${\mathcal {A}}_{n,\lambda }(x)$.

Theorem 5

For any partition $\lambda$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned} {\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,\frac{1}{1-x}=\frac{{\mathcal {A}}_{n,\lambda ^{(n)}}(x)}{(1-x)^{n+1}}. \end{aligned}$$

(15)

Proof

By (14) we obtain

$$\begin{aligned} {\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,\frac{1}{1-x}&=\sum _{i\ge 0}r_i(\lambda )\,{\mathbf {X}}^{\ell (\lambda )-i}{\mathbf {D}}^{n-i}\,\frac{1}{1-x}\\&=\sum _{i\ge 0}r_i(\lambda )\,\big (n-i\big )!\,\frac{x^{\ell (\lambda )-i}}{(1-x)^{n-i+1}}, \end{aligned}$$

hence

$$\begin{aligned} (1-x)^{n+1}{\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,\frac{1}{1-x}=\sum _{i\ge 0}r_i(\lambda )\,\big (n-i\big )!\,x^{\ell (\lambda )-i}\,(1-x)^i. \end{aligned}$$

(16)

Moreover, by (9) we have

$$\begin{aligned} x^{\ell (\lambda )}\,{\mathcal {A}}_{n,\lambda }(1/x)=\sum _{i\ge 0}r_i(\lambda )\,(n-i)!\,x^{\ell (\lambda )-i}(1-x)^i. \end{aligned}$$

(17)

Finally, by comparing (17), (16) and Proposition 1 (iii), we have

$$\begin{aligned}\frac{{\mathcal {A}}_{n,\lambda ^{(n)}}(x)}{(1-x)^{n+1}}=\frac{x^{\ell (\lambda )}{\mathcal {A}}_{n,\lambda }(1/x)}{(1-x)^{n+1}}={\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,\frac{1}{1-x}.\end{aligned}$$

$\square$

A first consequence of (15) is the following extension of the Worpitzky identity for Eulerian polynomials.

Corollary 6

Let m be a positive integer. For any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned} \prod _{i=0}^{n-1}(m+\lambda '_{n-i}-i)=\sum _{k\ge 0}\left( {\begin{array}{c}m+\ell (\lambda )-k\\ n\end{array}}\right) \,{\mathcal {A}}_{n,k,\lambda ^{(n)}}, \end{aligned}$$

(18)

where $\lambda '=(\lambda '_1,\lambda '_2,\ldots ,\lambda '_{l'})$ is the conjugate of $\lambda$ and we assume that $\lambda '_i=0$ for $i>l'=\lambda _1$.

Proof

Set

$$\begin{aligned}\mu :=(\underbrace{n,n,\ldots ,n}_{m},\lambda _1,\lambda _2,\ldots ,\lambda _l)\end{aligned}$$

and observe that

$$\begin{aligned}r_{n}(\mu )=\prod _{i=0}^{n-1}(m+\lambda '_{n-i}-i).\end{aligned}$$

Moreover, we have ${\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,x^m={\varvec{\Pi }}_{\mu }\,1=r_{n}(\mu )\,x^{m+\ell (\lambda )-n}$ and then the left-hand side of (15) is given by

$$\begin{aligned}{\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,\frac{1}{1-x}=\sum _{m\ge 0}\prod _{i=0}^{n-1}(m+\lambda '_{n-i}-i)\,x^{m+\ell (\lambda )-n}.\end{aligned}$$

From (6), the right-hand side of (15) may be rewritten as

$$\begin{aligned}\sum _{i\ge 0}\bigg (\sum _{k\ge 0}\left( {\begin{array}{c}n+i-k\\ n\end{array}}\right) \,{\mathcal {A}}_{n,k,\lambda ^{(n)}}\bigg )x^i.\end{aligned}$$

Hence, (18) follows by extracting the coefficient of $x^{m-n+\ell (\lambda )}$ from both sides in (15). $\square$

Example 3

Setting $\lambda =(n-1,n-2,\ldots ,r)$ in (18), and observing that $\lambda ^{(n)}=\delta _{n-r}$, we obtain the following Worpitzky identity [7],

$$\begin{aligned}m^{n-r}\,\frac{m!}{(m-r)!}=\sum _{k\ge 0}\left( {\begin{array}{c}m+r-k\\ n\end{array}}\right) \,^{r}\!A_{n,k}.\end{aligned}$$

Of course, $r=1$ leads to the Worpitzky identity for Eulerian numbers:

$$\begin{aligned}m^n=\sum _{k\ge 0}\left( {\begin{array}{c}m+1-k\\ n\end{array}}\right) \,A_{n,k}.\end{aligned}$$

A further consequence of (15) is a remarkable property of the polynomials ${\mathcal {A}}_{n,\lambda }(x)$ with respect to derivation. In terms of the underlined Young diagrams, this property encodes the evolution of the polynomials ${\mathcal {A}}_{n,\lambda }(x)$, for a fixed partition $\lambda$, with respect to square diagrams $D_n$ of increasing size.

Corollary 7

For any partition $\lambda$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned} {\mathbf {D}}\,\frac{{\mathcal {A}}_{n,\lambda }(x)}{(1-x)^{n+1}}=\frac{{\mathcal {A}}_{n+1,\lambda }(x)}{(1-x)^{n+2}} .\end{aligned}$$

(19)

Proof

If $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ then we set $\lambda +1:=(\lambda _1+1,\lambda _2+1,\ldots ,\lambda _l+1)$. Note that the reduced complements of $\lambda$ in $D_n$ and of $\lambda +1$ in $D_{n+1}$ agree, hence from (15) we have

$$\begin{aligned}{\mathbf {D}}\,\frac{{\mathcal {A}}_{n,\lambda ^{(n)}}(x)}{(1-x)^{n+1}}={\mathbf {D}}{\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{n-\lambda _1}\,\frac{1}{1-x}={\varvec{\Pi }}_{\lambda +1}{\mathbf {D}}^{(n+1)-(\lambda _1+1)}\,\frac{1}{1-x}=\frac{{\mathcal {A}}_{n+1,\lambda ^{(n)}}(x)}{(1-x)^{n+2}}.\end{aligned}$$

$\square$

Identity (19) suggests that the polynomials ${\mathcal {A}}_{n,\lambda }(x)$ indexed by the smallest n such that $D_\lambda \subseteq D_n$, play a special role. Indeed, for any partition $\lambda$, we set

$$\begin{aligned} n(\lambda ):= {\max }\{\lambda _1,\ell (\lambda )\} \end{aligned}$$

(20)

and define

$$\begin{aligned} {\mathcal {A}}_{\lambda }(x):={\mathcal {A}}_{n(\lambda ),\lambda }(x). \end{aligned}$$

(21)

Hence, we obtain the following recursive rule.

Corollary 8

For any partition $\lambda$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned} {\mathcal {A}}_{n,\lambda }(x)=(1-x)^{n+1}\,{\mathbf {D}}^{n-n(\lambda )}\,\frac{{\mathcal {A}}_{\lambda }(x)}{(1-x)^{n(\lambda )+1}}. \end{aligned}$$

(22)

Proof

Identity (22) follows by iterating (19). $\square$

Remark 1

Note that, by Proposition 1 (iii) and (7) we have ${\mathcal {A}}_{\delta _n}(x)=x\,A_{n}(x)$. Therefore, by setting $\lambda =\delta _{n-r}$ in (22), the polynomials ${}^r\!A_n(x)$ are obtained via suitable derivatives involving the classical Eulerian polynomials,

$$\begin{aligned}{}^r\!A_{n}(x)=(1-x)^{n+1}\,{\mathbf {D}}^{r}\,\frac{x\,A_{n-r}(x)}{(1-x)^{n-r+1}}.\end{aligned}$$

5 q-analogues arising from the q-Weyl algebra

Let ${\mathbf {D}}_q$ denote the q-derivative operator acting on the polynomial p(x) according to the following rule,

$$\begin{aligned}{\mathbf {D}}_q\,p(x)=\frac{p(qx)-p(x)}{qx-x}.\end{aligned}$$

We have ${\mathbf {D}}_q{\mathbf {X}}-q{\mathbf {X}}{\mathbf {D}}_q=1$ and the algebra generated by ${\mathbf {X}},{\mathbf {D}}_q$ is a q-analogue of the Weyl algebra. Now, let $[i]:=1+q+\cdots +q^{i-1}$ denote the q-integer, and for all partitions $\lambda$, let ${\varvec{\Pi }}_{\lambda ,q}$ be obtained from (13) by replacing ${\mathbf {D}}$ with ${\mathbf {D}}_q$. As ${\mathbf {D}}_q^i\,x^m=[m][m-1]\cdots [m-i+1]\,x^{m-i}$, straightforward computations show that

$$\begin{aligned} {\varvec{\Pi }}_{\lambda ,q}{\mathbf {D}}_{q}^{n-\lambda _1}\,\frac{1}{1-x}=\sum _{m\ge 0}\prod _{i=0}^{n-1}[m+\lambda '_{n-i}-i]\,x^{m-n+\ell (\lambda )}. \end{aligned}$$

(23)

Note that the right-hand side of (23) agrees with the right-hand side of identity (I.11) in the paper of Garsia and Remmel [8] , as can be seen by setting $a_{i+1}=n-\ell (\lambda )+\lambda '_{n-i}$ for $0\le i\le n-1$, that is by setting $\lambda =\mu ^{(n)}$ for $\mu :=(a_n,a_{n-1},\ldots ,a_1)$. Now, we let ${\mathcal {A}}_{n,\lambda ^{(n)}}(x,q)$ denote the polynomial defined by

$$\begin{aligned} {\varvec{\Pi }}_{\lambda ,q}{\mathbf {D}}_{q}^{n-\lambda _1}\,\frac{1}{1-x}=\frac{{\mathcal {A}}_{n,\lambda ^{(n)}}(x,q)}{(1-x)(1-xq)\cdots (1-xq^n)}, \end{aligned}$$

(24)

and the right-hand side of (I.12) in [8] ensures that $Q_A(x,q)={\mathcal {A}}_{n,\lambda ^{(n)}}(x,q)$ when the partition $\lambda$ is chosen such that $a_{i+1}=n-\ell (\lambda )+\lambda '_{n-i}$ for $0\le i\le n-1$. First, we recall that

$$\begin{aligned}\frac{1}{(1-x)(1-xq)\cdots (1-xq^n)}=\sum _{k\ge 0}\bigg [{\genfrac{}{}{0.0pt}{}{n+k}{n}}\bigg ]x^k.\end{aligned}$$

Moreover, we define ${\mathcal {A}}_{n,k,\lambda ^{(n)}}(q)$ by

$$\begin{aligned}{\mathcal {A}}_{n,\lambda ^{(n)}}(x,q)=\sum _{k\ge 0}{\mathcal {A}}_{n,k,\lambda ^{(n)}}(q)\,x^k,\end{aligned}$$

and compare the coefficients of (23) and (24) to obtain the following q-analogue of Corollary 6.

Corollary 9

Let m be a positive integer. For any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned}\prod _{i=0}^{n-1}[m+\lambda '_{n-i}-i]=\sum _{k\ge 0}\bigg [\genfrac{}{}{0.0pt}{}{m+\ell (\lambda )-k}{n}\bigg ]\,{\mathcal {A}}_{n,k,\lambda ^{(n)}}(q),\end{aligned}$$

where $\lambda '=(\lambda '_1,\lambda '_2,\ldots ,\lambda '_{l'})$ is the conjugate of $\lambda$ and we assume that $\lambda '_i=0$ for $i>l'=\lambda _1$.

Moreover, simply by replacing ${\mathbf {D}}$ with ${\mathbf {D}}_q$ in the proof of Corollary 7, we obtain the following q-analogue of (19).

Corollary 10

For any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned}{\mathbf {D}}_q\,\frac{{\mathcal {A}}_{n,\lambda }(x,q)}{(1-x)(1-xq)\cdots (1-xq^n)}=\frac{{\mathcal {A}}_{n+1,\lambda }(x,q)}{(1-x)(1-xq)\cdots (1-xq^{n+1})}.\end{aligned}$$

We let ${\mathcal {A}}_{\lambda }(x,q):={\mathcal {A}}_{n(\lambda ),\lambda }(x,q)$ and easily obtain the q-analogue of the recursive property (22).

Corollary 11

For any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ and any positive integer n such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned}{\mathcal {A}}_{n+1,\lambda }(x,q)=(1-x)(1-xq)\cdots (1-xq^{n+1}){\mathbf {D}}_q^{n-n(\lambda )}\,\frac{{\mathcal {A}}_{\lambda }(x,q)}{(1-x)(1-xq)\cdots (1-xq^{n(\lambda )})}.\end{aligned}$$

We explicitly remark that the polynomials ${\mathcal {A}}_{n,k,\lambda }(q)$ are the so-called q-hit numbers [5].

6 Further generalizations and applications

6.1 An application to the operator $({\mathbf {X}}^r{\mathbf {D}}^s)^n$

We now consider the polynomials $A_{r,s,n}(x)$ introduced in [1] and defined by

$$\begin{aligned}({\mathbf {X}}^r{\mathbf {D}}^s)^n\,\frac{1}{1-x}=\frac{A_{r,s,n}(x)}{(1-x)^{sn+1}},\end{aligned}$$

for all positive integers $r\le s$ and $n\ge 1$. Let ${\varvec{r}}=(r_1,r_2,\ldots ,r_{n})$ and ${\varvec{u}}=(u_1,u_2,\ldots ,u_{n})$ satisfy $r_1=r_2=\ldots =r_{n}=s$ and $u_1=u_2=\ldots =u_{n}=r$, set $\delta _{r,s,n}:=\lambda _{{\varvec{r}},{\varvec{u}}}$. The Young diagram of $\delta _{r,s,n}$ is obtained from $D_{\delta _n}$ by replacing each box in $D_{\delta _n}$ with a rectangular diagram of s columns and r rows. For example, the Young diagram of $\delta _{2,3,2}$ is $D_{(6,6,3,3)}$ , as shown in Fig. 3 (dark gray) as a subset of $D_9$. We denote by $\mathrm {exc}_{r,s,n}$ the deformation of the excedance statistic induced by $\lambda =\delta _{r,s,n}$ via (11). In particular, for all $\sigma \in {\mathfrak {S}}_{sn}$, we have

$$\begin{aligned} \mathrm {exc}_{r,s,n-1}(\sigma )=\big |\{i=(i_1-1)r+i_2\,|\,1\le i_1\le n-1,\,1\le i_2\le r,\,\sigma _{i}>si_1)\}\big |.\nonumber \\ \end{aligned}$$

(25)

Note that, as $\delta _{1,1,n-1}=\delta _{n-1}$ (by convention $\delta _0=(1)$), we have $\mathrm {exc}_{1,1,n-1}(\sigma )=\mathrm {exc}(\sigma )$ for all $\sigma \in {\mathfrak {S}}_n$. The following result gives a combinatorial explanation for the identity $A_{r,s,n}(1)=(sn)!$ [1].

Proposition 12

For all positive integers $r\le s$ and $n\ge 1$, we have

$$\begin{aligned} A_{r,s,n}(x)=x^r{\mathcal {A}}_{sn,\delta _{r,s,n-1}}(x)=x^r\sum _{\sigma \in {\mathfrak {S}}_{sn}}x^{\mathrm {exc}_{r,s,n-1}(\sigma )}. \end{aligned}$$

(26)

Proof

Let $\lambda :=\delta _{s,r,n-1}$. From

$$\begin{aligned}({\mathbf {X}}^r{\mathbf {D}}^s)^n={\mathbf {X}}^r({\mathbf {D}}^s{\mathbf {X}}^r)^{n-1}{\mathbf {D}}^s={\mathbf {X}}^r{\varvec{\Pi }}_{\lambda }{\mathbf {D}}^{sn-s(n-1)},\end{aligned}$$

by virtue of Theorem 5 we obtain

$$\begin{aligned}\frac{A_{r,s,n}(x)}{(1-x)^{sn+1}}={\mathbf {X}}^r{\varvec{\Pi }}_\lambda {\mathbf {D}}^{sn-s(n-1)}\frac{1}{1-x}=\frac{x^r\,{\mathcal {A}}_{sn,\lambda ^{(sn)}}(x)}{(1-x)^{sn+1}}.\end{aligned}$$

As $\delta _{r,s,n-1}=\delta _{r,s,n-1}^{(sn)}$,

$$\begin{aligned}A_{r,s,n}(x)=x^r\,{\mathcal {A}}_{sn,\delta _{r,s,n-1}}(x),\end{aligned}$$

and via (12) we deduce (26). $\square$

Now, we prove the following result originally conjectured in [1].

Proposition 13

For all positive integers $r\le s$ and $n\ge 1$, we have

$$\begin{aligned}A_{r,s,n}(x)=x^{r(n-1)}A_{r,s,n}(1/x).\end{aligned}$$

Proof

By taking into account Proposition 1(iii), as $\delta _{r,s,n-1}=\delta _{r,s,n-1}^{(sn)}$, and since $\ell (\delta _{r,s,n-1})=r(n-1)$, from (26), we have

$$\begin{aligned}x^{r(n-1)}\,A_{r,s,n}(1/x)=x^r\,x^{\ell (\delta _{r,s,n-1})}{\mathcal {A}}_{sn,\delta _{r,s,n-1}}(1/x)=A_{r,s,n}(x).\end{aligned}$$

$\square$

6.2 Generalizations of the Dobiński formula

One may think to replace the geometric series $1/(1-x)$ in (15) and let any product ${\varvec{\Pi }}$ act on an arbitrary power series f(x). More interestingly, one may look for those series f(x) such that ${\varvec{\Pi }}\,f(x)$ has some combinatorial interest. Let us discuss the case $f(x)=e^x$, which leads to an extension of the Dobiński formula. Indeed, by (14) one obtains

$$\begin{aligned}{\varvec{\Pi }}_\lambda {\mathbf {D}}^{n-\lambda _1}\,e^x=e^x\sum _{k\ge 0}r_k(\lambda )\,x^{\ell (\lambda )-k}=e^x\,x^{\ell (\lambda )}\,R_{\lambda }(1/x),\end{aligned}$$

where $R_\lambda (x)=\sum _{k}\,r_k(\lambda )\,x^k$ is the well-known rook polynomial associated with $D_\lambda$. On the other hand, by expanding $e^x$ we also have

$$\begin{aligned}{\varvec{\Pi }}_\lambda {\mathbf {D}}^{n-\lambda _1}\,e^x=\sum _{m\ge 0}\prod _{i=0}^{n-1}(m+\lambda '_{n-i}-i)\,\frac{x^{m-n+\ell (\lambda )}}{m!},\end{aligned}$$

and then

$$\begin{aligned}\sum _{m\ge 0}\prod _{i=0}^{n-1}(m+\lambda '_{n-i}-i)\,\frac{x^{m-n+\ell (\lambda )}}{m!}=e^x\,x^{\ell (\lambda )}\,R_{\lambda }(1/x).\end{aligned}$$

Setting $x=1$ and $R_\lambda :=R_\lambda (1)$ we obtain the following generalization of the Dobiński formula

$$\begin{aligned} \sum _{m\ge 0}\frac{\prod _{i=0}^{n-1}(m+\lambda '_{n-i}-i)}{m!}=e\,R_{\lambda }. \end{aligned}$$

(27)

The classical case arises when $\lambda =\delta _{n-1}$, and then $R_{\delta _{n-1}}=B_n$ is the n-th Bell number,

$$\begin{aligned}\sum _{m\ge 0}\frac{m^n}{m!}=e\,B_n.\end{aligned}$$

Moreover, replacing n with sn, setting $\lambda =\delta _{r,s,n-1}$ and $B_{r,s,n}:=R_{\delta _{r,s,n-1}}$, we get a Dobiński formula for the sum of all generalized Stirling numbers $S_{r,s}(n,k):=r_{sn-k}(\delta _{r,s,n-1})$ [2],

$$\begin{aligned}\sum _{m\ge 0}\frac{1}{\big (m-(s-r)n\big )!}\prod _{i=1}^{n}\frac{\big (m-(s-r)i\big )!}{\big (m-(s-r)i-r\big )!}=e\,B_{r,s,n}.\end{aligned}$$

In particular, when $r=s$, we recover

$$\begin{aligned}\sum _{m\ge 0}\frac{1}{m!}\frac{m!^{n}}{(m-r)!^n}=e\,B_{r,r,n}.\end{aligned}$$

In closing, to recover a q-analogue of (27), set

$$\begin{aligned}\varepsilon (x):=\sum _{k\ge 0}\frac{x^k}{[k]!},\end{aligned}$$

where $[k]!:=[1][2]\cdots [k]$, and observe that ${\mathbf {D}}_q\,\varepsilon (x)=\varepsilon (x)$. We deduce

$$\begin{aligned}{\varvec{\Pi }}_{\lambda ,q}{\mathbf {D}}_q^{n-\lambda _1}\,\varepsilon (x)=\varepsilon (x)\sum _{k\ge 0}r_k(\lambda ,q)\,x^{\ell (\lambda )-k}=\varepsilon (x)\,x^{\ell (\lambda )}\,R_{\lambda }(1/x,q),\end{aligned}$$

where $R_\lambda (x,q)=\sum _{k}r_k(\lambda ,q)\,x^{\ell (\lambda )-k}$, and the $r_k(\lambda ,q)$ are the q-rook numbers arising here as the normal ordering coefficients of ${\varvec{\Pi }}_{\lambda ,q}{\mathbf {D}}_q^{n-\lambda _1}$ (Theorem 6.1 in [17]). Finally, we set $\varepsilon :=\varepsilon (1)$ and $R_\lambda (q):=R_\lambda (1,q)$ and obtain the following result.

Proposition 14

For any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ such that $D_\lambda \subseteq D_n$, we have

$$\begin{aligned} \sum _{m\ge 0}\frac{\prod _{i=0}^{n-1}[m+\lambda '_{n-i}-i]}{[m]!}=\varepsilon \,R_{\lambda }(q), \end{aligned}$$

(28)

where $\lambda '=(\lambda '_1,\lambda '_2,\ldots ,\lambda '_{l'})$ is the conjugate of $\lambda$ and we assume that $\lambda '_i=0$ for $i>l'=\lambda _1$.