{
"id": "1702.06666",
"version": "v1",
"published": "2017-02-22T03:44:09.000Z",
"updated": "2017-02-22T03:44:09.000Z",
"title": "Gamma-positivity of variations of Eulerian polynomials",
"authors": [
"John Shareshian",
"Michelle L. Wachs"
],
"comment": "29 pages",
"categories": [
"math.CO"
],
"abstract": "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 Sch\\\"utzenberger shows that the Eulerian polynomials have a stronger property, namely $\\gamma$-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-Sch\\\"utzenberger 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.",
"revisions": [
{
"version": "v1",
"updated": "2017-02-22T03:44:09.000Z"
}
],
"analyses": {
"subjects": [
"05A05",
"05E05",
"05E10",
"05E45",
"52B05"
],
"keywords": [
"gamma-positivity",
"variations",
"binomial-eulerian polynomials",
"analogous symmetric function identities",
"symmetric function analogs"
],
"note": {
"typesetting": "TeX",
"pages": 29,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}