The Real Stability of the Multivariate Eulerian Polynomials for Restricted Excedance Statistics

Abstract

In this paper, we construct multivariate generalizations of the Eulerian polynomials, the type B analogues and the r-colored Eulerian polynomials for restricted excedance statistics by using the context-free grammars. Furthermore, the grammars enable us to obtain the combinatorial interpretations and the recurrence relations for these multivariate polynomials in terms of permutations on the symmetric group, the hyperoctahedral group, the wreath product of the cyclic group and the symmetric group. Based on a sufficient condition for the real stability of recursive multivariate sequences of polynomials, we obtain the real stability of these multivariate polynomials. As applications, we show the real-rootedness of the univariate Eulerian polynomials, the type B analogues and the r-colored Eulerian polynomials for restricted excedance statistics in a unified manner.

Keywords

Real stability / real-rooted polynomials / excedance / context-free grammar / multivariate generalizations of Eulerian polynomials / 05A05 / 32A60 / 68Q42

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Peking University