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arXiv · 2006.14064

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

G. -N. Han·S. -M. Ma
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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, Andr\'e 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.

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G. -N. Han, S. -M. Ma. 2020-06-24. Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux. https://arxiv.org/abs/2006.14064

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