Gamma-positivity of Variations of Eulerian Polynomials
نویسندگان
چکیده
An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are h-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Schützenberger shows that the Eulerian polynomials have a stronger property, namely γ-positivity, and a formula of Postnikov, Reiner and Williams does the same for the binomial-Eulerian polynomials. We obtain q-analogs of both the Foata-Schützenberger formula and an alternative to the Postnikov-Reiner-Williams formula, and we show that these q-analogs are specializations of analogous symmetric function identities. Algebro-geometric interpretations of these symmetric function analogs are presented.
منابع مشابه
ar X iv : 1 71 1 . 05 98 3 v 2 [ m at h . C O ] 1 9 N ov 2 01 7 GAMMA - POSITIVITY IN COMBINATORICS AND GEOMETRY
Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Schützenberger and Strehl on the Eulerian polynomials; it was revived independently by Brändén and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, r...
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