Gamma-positivity of variations of Eulerian polynomials
| Autor: | Michelle L. Wachs, John Shareshian |
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| Rok vydání: | 2017 |
| Předmět: |
Class (set theory)
Property (philosophy) Mathematics::Combinatorics 010102 general mathematics Eulerian path Polytope 0102 computer and information sciences 16. Peace & justice 01 natural sciences Interpretation (model theory) Combinatorics Symmetric function Identity (mathematics) symbols.namesake 010201 computation theory & mathematics symbols FOS: Mathematics Mathematics - Combinatorics 05A05 05E05 05E10 05E45 52B05 Combinatorics (math.CO) 0101 mathematics Binomial coefficient Mathematics |
| DOI: | 10.48550/arxiv.1702.06666 |
| Popis: | 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. Comment: 30 pages; v2: typo in equation (1.5) correcte; v3: further minor corrections and further discussion of equivariant Gal's phenomenon |
| Databáze: | OpenAIRE |
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