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 qderivative operator, we extend these results to the qanalogue setting including the qhit numbers. Finally, a combinatorial description and a proof of the symmetry of a family of polynomials introduced by one of the authors are provided.
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 nonattacking 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 nonattacking rook placements P on D with \(P=k\) is said to be the kth rook number of D. If \(D=D_{\lambda }\) is the Young diagram of a partition \(\lambda \), then we write \(r_k(\lambda )\) for the kth rook number of \(D_\lambda \). In particular, for the staircase partition \(\delta _{n}:=(n,n1,\ldots ,1)\), it is wellknown that the rook numbers \(r_k(\delta _{n1})\) are the Stirling numbers of the second kind \(S(n,nk)\). 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
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 wellknown that the classical Eulerian polynomials \(A_n(x)\) arise as \({\mathcal {A}}_{n,\delta _{n1}}(x)\). In Sect. 3, we will show that \({\mathcal {A}}_{n,\delta _{nr}}(x)\) 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]
Based on a qanalogue of rook numbers, Garsia and Remmel [8] provided a qanalogue 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 qhit 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 socalled 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
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 \(1/(1x)\). 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 \(D_\lambda \),
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 qderivative \({\mathbf {D}}_q\), acts on \(1/(1x)\). More precisely, they are the polynomials \({\mathcal {A}}_{n,\lambda }(x,q)\) such that
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 \(\lambda =\delta _{nr}\) 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 qanalogue 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 qanalogue (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
and introduced by one of the authors of the present paper [1].
Partitions and rook numbers
By a partition, we mean a finite nonincreasing 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 leftaligned 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 ith row (counting from top to bottom) and at the jth 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.
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.
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 \(\lambda ^{(n)}:=(n\lambda _l,n\lambda _{l1},\ldots ,n\lambda _1)\). 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.
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 kth 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.
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
Moreover, given \(\sigma \in {\mathfrak {S}}_n\), let \(\sigma ^\lambda =\sigma ^\lambda _1\sigma ^\lambda _2\ldots \sigma ^\lambda _n\) be defined by
It is easy to deduce that \(\sigma \mapsto \sigma ^\lambda \) is a bijective map. Now, set
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.
As \(\big \sigma \cap A_\lambda \big =\ell (\lambda )\), we obtain
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:
Moreover, we set
and obtain
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
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
and
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
Moreover, if \((\lambda ')^{(n)}=\lambda '\) then
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 \(\{(x1)^i\,\,i\ge 0\}\) 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
Proof
By (5) we have
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 \(ni\) available boxes on the main diagonal of \(D:=D_n\), that is
Clearly \((\sigma ^{B_0},B_0)\in \mathrm {Pairs}\). Moreover, we obtain all the pairs of type \((\sigma ,B_0)\) by permuting the \(ni\) 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 \(r_i(\lambda )\,(ni)!\). We recover
which gives (9) when x is replaced by \(x1\). \(\square \)
Example 2
Let r be a nonnegative integer. Following Foata and Schützenberger [7], we consider the polynomial
where
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 \(\sigma _i\le n+1i\) if and only if \(\sigma '_i\ge r+i\). As a consequence, we obtain
or equivalently \({}^r\!\!A_{n}(x)={\mathcal {A}}_{n,\delta _{nr}}(x)\). From (9), we recover the following Frobenius identity for the polynomials \({}^r\!\!A_{n}(x)\) [7]:
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
where \(\lambda _i=0\) is assumed for \(\ell (\lambda )<i\le n\). As before, the complement bijection \(\sigma \mapsto \sigma '\) provides
so that we get
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
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
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
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
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,
Then, we conclude
\(\square \)
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
Proof
By (14) we obtain
hence
Moreover, by (9) we have
Finally, by comparing (17), (16) and Proposition 1 (iii), we have
\(\square \)
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
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
and observe that
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 lefthand side of (15) is given by
From (6), the righthand side of (15) may be rewritten as
Hence, (18) follows by extracting the coefficient of \(x^{mn+\ell (\lambda )}\) from both sides in (15). \(\square \)
Example 3
Setting \(\lambda =(n1,n2,\ldots ,r)\) in (18), and observing that \(\lambda ^{(n)}=\delta _{nr}\), we obtain the following Worpitzky identity [7],
Of course, \(r=1\) leads to the Worpitzky identity for Eulerian numbers:
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
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
\(\square \)
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
and define
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
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 \(\lambda =\delta _{nr}\) in (22), the polynomials \({}^r\!A_n(x)\) are obtained via suitable derivatives involving the classical Eulerian polynomials,
qanalogues arising from the qWeyl algebra
Let \({\mathbf {D}}_q\) denote the qderivative operator acting on the polynomial p(x) according to the following rule,
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 qanalogue of the Weyl algebra. Now, let \([i]:=1+q+\cdots +q^{i1}\) denote the qinteger, and for all partitions \(\lambda \), let \({\varvec{\Pi }}_{\lambda ,q}\) be obtained from (13) by replacing \({\mathbf {D}}\) with \({\mathbf {D}}_q\). As \({\mathbf {D}}_q^i\,x^m=[m][m1]\cdots [mi+1]\,x^{mi}\), straightforward computations show that
Note that the righthand side of (23) agrees with the righthand side of identity (I.11) in the paper of Garsia and Remmel [8] , as can be seen by setting \(a_{i+1}=n\ell (\lambda )+\lambda '_{ni}\) for \(0\le i\le n1\), that is by setting \(\lambda =\mu ^{(n)}\) for \(\mu :=(a_n,a_{n1},\ldots ,a_1)\). Now, we let \({\mathcal {A}}_{n,\lambda ^{(n)}}(x,q)\) denote the polynomial defined by
and the righthand 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 \(a_{i+1}=n\ell (\lambda )+\lambda '_{ni}\) for \(0\le i\le n1\). First, we recall that
Moreover, we define \({\mathcal {A}}_{n,k,\lambda ^{(n)}}(q)\) by
and compare the coefficients of (23) and (24) to obtain the following qanalogue 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
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 qanalogue 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
We let \({\mathcal {A}}_{\lambda }(x,q):={\mathcal {A}}_{n(\lambda ),\lambda }(x,q)\) and easily obtain the qanalogue 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
We explicitly remark that the polynomials \({\mathcal {A}}_{n,k,\lambda }(q)\) are the socalled qhit numbers [5].
Further generalizations and applications
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
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
Note that, as \(\delta _{1,1,n1}=\delta _{n1}\) (by convention \(\delta _0=(1)\)), we have \(\mathrm {exc}_{1,1,n1}(\sigma )=\mathrm {exc}(\sigma )\) 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
Proof
Let \(\lambda :=\delta _{s,r,n1}\). From
by virtue of Theorem 5 we obtain
As \(\delta _{r,s,n1}=\delta _{r,s,n1}^{(sn)}\),
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
Proof
By taking into account Proposition 1(iii), as \(\delta _{r,s,n1}=\delta _{r,s,n1}^{(sn)}\), and since \(\ell (\delta _{r,s,n1})=r(n1)\), from (26), we have
\(\square \)
Generalizations of the Dobiński formula
One may think to replace the geometric series \(1/(1x)\) 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
where \(R_\lambda (x)=\sum _{k}\,r_k(\lambda )\,x^k\) is the wellknown rook polynomial associated with \(D_\lambda \). On the other hand, by expanding \(e^x\) we also have
and then
Setting \(x=1\) and \(R_\lambda :=R_\lambda (1)\) we obtain the following generalization of the Dobiński formula
The classical case arises when \(\lambda =\delta _{n1}\), and then \(R_{\delta _{n1}}=B_n\) is the nth Bell number,
Moreover, replacing n with sn, setting \(\lambda =\delta _{r,s,n1}\) and \(B_{r,s,n}:=R_{\delta _{r,s,n1}}\), we get a Dobiński formula for the sum of all generalized Stirling numbers \(S_{r,s}(n,k):=r_{snk}(\delta _{r,s,n1})\) [2],
In particular, when \(r=s\), we recover
In closing, to recover a qanalogue of (27), set
where \([k]!:=[1][2]\cdots [k]\), and observe that \({\mathbf {D}}_q\,\varepsilon (x)=\varepsilon (x)\). We deduce
where \(R_\lambda (x,q)=\sum _{k}r_k(\lambda ,q)\,x^{\ell (\lambda )k}\), and the \(r_k(\lambda ,q)\) are the qrook 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
where \(\lambda '=(\lambda '_1,\lambda '_2,\ldots ,\lambda '_{l'})\) is the conjugate of \(\lambda \) and we assume that \(\lambda '_i=0\) for \(i>l'=\lambda _1\).
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Acknowledgements
The first and the fourth authors were partially supported by FCTFundação para a Ciência e a Tecnologia (Portugal), through the project UIDB/04721/2020.
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Agapito, J., Petrullo, P., Senato, D. et al. Eulerian polynomials via the Weyl algebra action. J Algebr Comb (2020). https://doi.org/10.1007/s10801020009976
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Keywords
 Eulerian polynomials
 Weyl algebra
 Rook numbers
 Permutation statistics
 Formal power series