Eulerian polynomials via the Weyl algebra action
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Eulerian polynomials via the Weyl algebra action Jose Agapito2 · Pasquale Petrullo1
· Domenico Senato1 · Maria M. Torres2
Received: 17 March 2020 / Accepted: 16 November 2020 © The Author(s) 2020
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 Dobi´nski 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. Keywords Eulerian polynomials · Weyl algebra · Rook numbers · Permutation statistics · Formal power series
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
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Pasquale Petrullo [email protected] Jose Agapito [email protected] Domenico Senato [email protected] Maria M. Torres [email protected]
1
Dipartimento di Scienze Umane, Università degli Studi della Basilicata, Via Nazario Sauro 85, Potenza, Italy
2
Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Campo Grande, Lisboa, Portugal
123
Journal of Algebraic Combinatorics
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 rk (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 = Dλ is the Young diagram of a partition λ, then we write rk (λ) for the k-th rook number of Dλ . In particular, for the staircase partition δn := (n, n − 1, . . . , 1), it is well-known that the rook numbers rk (δn−1 ) are the Stirling numbers of the second kind S(n, n − k). In this sense, the sum Rλ = k rk (λ) can be regarded as a generalized Bell number. By identifying the permutations in the symmetric group Sn with the placements on the square diagram Dn consisting of n rows of length n, for any partition λ such that Dλ ⊆ Dn , we set An,λ (x) :=
x |σ ∩Dλ | .
σ ∈Sn
The polynomials An,λ (x) often occur within the well developed literature on rook theory [4,6,9–14]. It is well-known that the classical Eulerian polynomials An (x) arise as An,δn−1 (x). In Sect. 3, we will show that An,δn−r (x) agrees with the polynomial r An (x) in
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