Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Abstract

In this paper we first present summation formulas for k-order Eulerian polynomials and 1/k-Eulerian polynomials. We then present combinatorial expansions of (c(x)D) n in terms of inversion sequences as well as k-Young tableaux, where c(x) is a differentiable function in the indeterminate x and D is the derivative with respect to x. We define the g-indexes of k-Young tableaux and Young tableaux, which have important applications in combinatorics.By establishing some relations between k-Young tableaux and standard Young tableaux, we express Eulerian polynomials, second-order Eulerian polynomials, André polynomials and the generating polynomials of gamma coefficients of Eulerian polynomials in terms of standard Young tableaux, which imply a deep connection among these polynomials.

A Context-free Grammar for the $e$-Positivity of the Trivariate Second-order Eulerian Polynomials

“…Examining the factor of two in the grammar H, we are guided precisely to the structure of 0-1-2 increasing plane trees along with a natural grammatical labeling. This formulation is in agreement with the known interpretation of binary increasing trees on [n] with exactly k leaves and no vertices with left children only, as numerically described by Han-Ma [16] in terms of 0-1-2 increasing trees [n] with k leaves. We find it convenient to work with 0-1-2 increasing plane trees in order to describe the labeling consistent with the grammar H.…”

“…The nonnegativity of the coefficients γ n, k has been referred to as the γpositivity. This property of the Eulerian polynomials and other polynomials along with the q-analogues has been extensively studied ever since, see, for example, [1,3,7,10,11,13,16,18,19,20,21,22,23,24,25,26,29,30], to name a few.…”