Eulerian polynomials via the Weyl algebra action

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.


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 λ is the Young diagram of a partition λ, then we write r k (λ) for the k-th rook number of D λ . In particular, for the staircase partition δ n := (n, n − 1, . . . , 1), it is well-known that the rook numbers r k (δ n−1 ) are the Stirling numbers of the second kind S(n, n − k). In this sense, the sum R λ = k r k (λ) can be regarded as a generalized Bell number. By identifying the permutations in the symmetric group S n with the placements on the square diagram D n consisting of n rows of length n, for any partition λ such that D λ ⊆ D n , we set The polynomials A n,λ (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 A n,δ n−1 (x). In Sect. 3, we will show that A n,δ 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 A n,λ (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] A n,λ (x) = k≥0 r k (λ) (n − k)! (x − 1) k . (1) Based on a q-analogue of rook numbers, Garsia and Remmel [8] provided a q-analogue for the polynomials A n,λ (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 X denotes the operator of multiplication by x, and D = d dx denotes the usual derivative operator, then DX − XD = 1 and the algebra generated by X and D is referred to as the Weyl algebra. The normal ordering of any product involving a occurrences of the operator X and b occurrences of the operator D is given by where λ is a suitable partition associated with . In this setting, the Stirling numbers of the second kind arise as the normal ordering coefficients of = (XD) n .
We show that the polynomials A n,λ (x) naturally describe the action of any product of the operators D and X on the geometric series 1/(1 − x). More precisely, given a partition λ = (λ 1 , λ 2 , . . . , λ l ), we define an operator λ such that for any square diagram D n containing D λ , where λ (n) is a partition that we call the reduced complement of λ 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 λ,q D n−λ 1 q , obtained from λ D n−λ 1 by replacing D with the q-derivative D q , acts on 1/(1−x). More precisely, they are the polynomials A n,λ (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 A n,λ (x) with respect to derivation (Corollary 7), and a recursion formula to compute A n,λ (x) (Corollary 8).
When λ = δ 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 D with D q (Corollaries 9,10,11). Furthermore, by letting λ D n−λ 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 and introduced by one of the authors of the present paper [1].

Partitions and rook numbers
By a partition, we mean a finite non-increasing vector λ = (λ 1 , λ 2 , . . . , λ l ) of positive integers called parts of λ. The number of parts of λ is called the length of λ, and denoted by (λ). The Young diagram (or Ferrers board) of λ is a left-aligned array of boxes, displayed in (λ) rows consisting of λ 1 , λ 2 , . . . , λ 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 λ = (4, 4, 4, 2, 2, 1) is shown in Fig. 1A, with a bullet drawn in the box D 3,2 . The conjugate of λ is the partition λ whose diagram D λ is obtained by reflecting D λ with respect to its main diagonal. For example, the conjugate of λ = (4, 4, 4, 2, 2, 1) is λ = (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 two partitions λ and μ, we write λ ⊆ μ to mean that D λ ⊆ D μ . Moreover, we let D n denote the square Young diagram of n rows, and for any partition is obtained from D n by removing the boxes of D λ , deleting all the rows of D n lying below D λ , then rotating by 180 • . 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 λ consisting of k boxes, usually called the k-th rook number of λ, will be denoted by r k (λ). 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 S n , we will consider the permutation σ = σ 1 σ 2 . . . σ n and the placement {D 1,σ (1) , D 2,σ (2) , . . . , D n,σ (n) } on D = D n as the same object. For instance, we identify the permutations 123, 132, 213, 231, 312, 321 in S 3 with the following placements on D 3 : Note that σ −1 is obtained by reflecting σ in the main diagonal of D n . Hence, for all σ ∈ S n and for all λ such that D λ ⊆ D n we have Moreover, given σ ∈ S n , let σ λ = σ λ 1 σ λ 2 . . . σ λ n be defined by It is easy to deduce that σ → σ λ is a bijective map. Now, set Observe that σ λ is obtained by separately rotating by 180 • the rectangles A λ and B λ (with respect to their center). For instance, let λ = (2, 2, 1), n = 5 and σ = 13425, then we have σ λ = 23514 as depicted in Fig. 5.

Generalized Eulerian polynomials
Given a partition λ, and a positive integer n such that D λ ⊆ D n , we define the polynomial A n,λ (x) as follows: Moreover, we set and obtain We get 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 σ → σ −1 ) one obtains

Proposition 1 Given a partition λ and a positive integer n such that D
Proof From (5) and (2), we have (i) and (ii), respectively. Moreover, by means of σ → σ λ and (4) we have which gives (i).
An explicit expansion of A n,λ (x) in terms of the basis {(x − 1) i | i ≥ 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 λ and a positive integer n such that D λ ⊆ D n , we have
Proof By (5) we have where Pairs denotes the set of all (σ, B) such that σ ∈ S n and B ⊆ (σ ∩ D λ ). Note that for all (σ, B) ∈ Pairs, B is a placement on D λ . Now, for any given placement We recover which gives (9) when x is replaced by x − 1.

Example 2
Let r be a nonnegative integer. Following Foata and Schützenberger [7], we consider the polynomial Clearly, 1 A n (x) is the classical Eulerian polynomial. Now, let σ → σ denote the bijection defined on S n+r by σ i : or equivalently r A n (x) = A n,δ n−r (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 λ = (λ 1 , λ 2 , . . . , λ l ), a positive integer n such that D λ ⊆ D n , and a permutation σ = σ 1 σ 2 . . . σ n ∈ S n , we set where λ i = 0 is assumed for (λ) < i ≤ n. As before, the complement bijection σ → σ provides so that we get

The Weyl algebra action
Let D, X : Z[x] → Z[x] denote the derivative operator and the operator of multiplication by x, respectively. As DX − XD = 1 the following normal ordering problem may be posed: given any product involving a occurrences of the operator D and b occurrences of the operator X, find the coefficients c i ( ) 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 λ, we set where r = (r 1 , r 2 , . . . , r k ) and u = (u 1 , u 2 , . . . , u k ) are the unique vectors satisfying λ = λ r,u . Note that λ 1 = r 1 + r 2 + · · · + r k and (λ) = u 1 + u 2 + · · · + u k .

Theorem 5 For any partition λ and any positive integer n such that D λ ⊆ D n , we have
Proof By (14) we obtain Moreover, by (9) we have Finally, by comparing (17), (16) and Proposition 1 (iii), we have A first consequence of (15) is the following extension of the Worpitzky identity for Eulerian polynomials.

Proof
Moreover, we have λ D n−λ 1 x m = μ 1 = r n (μ) x m+ (λ)−n and then the left-hand side of (15) is given by From (6), the right-hand side of (15) may be rewritten as Hence, (18) follows by extracting the coefficient of x m−n+ (λ) from both sides in (15). (18), and observing that λ (n) = δ n−r , 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 A n,λ (x) with respect to derivation. In terms of the underlined Young diagrams, this property encodes the evolution of the polynomials A n,λ (x), for a fixed partition λ, with respect to square diagrams D n of increasing size.

Corollary 7 For any partition λ and any positive integer n such that D λ ⊆ D n , we have
Proof If λ = (λ 1 , λ 2 , . . . , λ l ) then we set λ + 1 := (λ 1 + 1, λ 2 + 1, . . . , λ l + 1). Note that the reduced complements of λ in D n and of λ + 1 in D n+1 agree, hence from (15) we have Identity (19) suggests that the polynomials A n,λ (x) indexed by the smallest n such that D λ ⊆ D n , play a special role. Indeed, for any partition λ, we set and define Hence, we obtain the following recursive rule.

Corollary 8 For any partition λ and any positive integer n such that D λ ⊆ D n , we have
Proof Identity (22) follows by iterating (19). (7) we have A δ n (x) = x A n (x). Therefore, by setting λ = δ n−r in (22), the polynomials r A n (x) are obtained via suitable derivatives involving the classical Eulerian polynomials,

q-analogues arising from the q-Weyl algebra
Let D q denote the q-derivative operator acting on the polynomial p(x) according to the following rule, We have D q X − qXD q = 1 and the algebra generated by X, D q is a q-analogue of the Weyl algebra. Now, let [i] := 1 + q + · · · + q i−1 denote the q-integer, and for all partitions λ, let λ,q be obtained from (13) by replacing D with D q . As 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 − (λ) + λ n−i for 0 ≤ i ≤ n − 1, that is by setting λ = μ (n) for μ := (a n , a n−1 , . . . , a 1 ). Now, we let A n,λ (n) (x, q) denote the polynomial defined by and the right-hand side of (I.12) in [8] ensures that Q A (x, q) = A n,λ (n) (x, q) when the partition λ is chosen such that a i+1 = n − (λ) + λ n−i for 0 ≤ i ≤ n − 1. First, we recall that and compare the coefficients of (23) and (24) to obtain the following q-analogue of Corollary 6.
Moreover, simply by replacing D with D q in the proof of Corollary 7, we obtain the following q-analogue of (19).
We explicitly remark that the polynomials A n,k,λ (q) are the so-called q-hit numbers [5].

An application to the operator (X r D s ) n
We now consider the polynomials A r ,s,n (x) introduced in [1] and defined by for all positive integers r ≤ s and n ≥ 1. Let r = (r 1 , r 2 , . . . , r n ) and u = (u 1 , u 2 , . . . , u n ) satisfy r 1 = r 2 = . . . = r n = s and u 1 = u 2 = . . . = u n = r , set δ r ,s,n := λ r,u . The Young diagram of δ r ,s,n is obtained from D δ n by replacing each box in D δ n with a rectangular diagram of s columns and r rows. For example, the Young diagram of δ 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 exc r ,s,n the deformation of the excedance statistic induced by λ = δ r ,s,n via (11). In particular, for all σ ∈ S sn , we have Note that, as δ 1,1,n−1 = δ n−1 (by convention δ 0 = (1)), we have exc 1,1,n−1 (σ ) = exc(σ ) for all σ ∈ S n . The following result gives a combinatorial explanation for the identity A r ,s,n (1) = (sn)! [1].

Generalizations of the Dobiński formula
One may think to replace the geometric series 1/(1 − x) in (15) and let any product act on an arbitrary power series f (x). More interestingly, one may look for those series f (x) such that 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 λ (x) = k r k (λ) x k is the well-known rook polynomial associated with D λ . On the other hand, by expanding e x we also have Setting x = 1 and R λ := R λ (1) we obtain the following generalization of the Dobiński formula The classical case arises when λ = δ n−1 , and then R δ n−1 = B n is the n-th Bell number, m≥0 m n m! = e B n .
Moreover, replacing n with sn, setting λ = δ r ,s,n−1 and B r ,s,n := R δ 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 (δ r ,s,n−1 ) [2], = e B r ,s,n .
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