Eulerian polynomials and the 𝑔-indices of Young tableaux

Abstract

In this paper, we introduce k k -Young tableaux and their g g -indices. We first present certain expansions of ( c ( x ) D ) n (c(x)D)^n in terms of inversion sequences as well as k k -Young tableaux, where c ( x ) c(x) is a smooth function in the indeterminate x x and D D is the derivative with respect to x x . By studying the connections between k k -Young tableaux and standard Young tableaux, we then present combinatorial interpretations of Eulerian polynomials, second-order Eulerian polynomials, and André polynomials in terms of standard Young tableaux.