Some combinatorial identities of the r-Whitney-Eulerian numbers

Toufik Mansour,Mark Shattuck,Sergio Villamaríín,José Ramírez

doi:10.2298/aadm180420012m

Abstract

In this paper, we study further properties of a recently introduced generalized Eulerian number, denoted by Am,r(n, k), which reduces to the classical Eulerian number when m = 1 and r = 0. Among our results is a generalization of an earlier symmetric Eulerian number identity of Chung, Graham and Knuth. Using the row generating function for Am,r(n, k) for a fixed n, we introduce the r-Whitney-Euler-Frobenius fractions, which generalize the Euler-Frobenius fractions. Finally, we consider a further four-parameter combinatorial generalization of Am,r(n, k) and find a formula for its exponential generating function in terms of the Lambert-W function.

Highlights

Recall that the r-Whitney numbers of the second kind Wm,r(n, k) are defined as connection constants in the polynomial identities n (1)(mx + r)n = mkWm,r(n, k)xk, n ≥ 0, k=0 where xn = x(x − 1) · · · (x − n + 1) if n ≥ 1 with x0 = 1
Given variables m and r, let Am,r(n, k) denote the r-Whitney-Eulerian numbers defined by Mezo and Ramırez [19] by the expression
If (m, r) = (1, r), we obtain the cumulative numbers studied by Dwyer [10, 11], see the Euler-Frobenius numbers considered by Gawronski and Neuschel [13]

Summary

Introduction

Recall that the r-Whitney numbers of the second kind Wm,r(n, k) (see, e.g., [17]) are defined as connection constants in the polynomial identities n (1). Note that in the final section of the current paper, we provide a combinatorial interpretation for a couple of polynomial generalizations of A(n, k) in terms of statistics on certain classes of permutations. We consider a further four-parameter polynomial generalization, denoted by A(n, k; a, b, c, d), of the Eulerian numbers and a related sequence. We provide a combinatorial interpretation for A(n, k; a, b, c, d) and an expression for the exponential generating function (e.g.f.) is found in terms of the Lambert-W function by solving explicitly a certain multi-parameter linear partial differential equation

A Special Quasi-Symmetric Identity

Identities Involving Combinatorial Sequences

Generalized Euler-Frobenius Fractions

Generalized Eulerian recurrences

A related distribution on Sn