Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

Autor: Hiranya Kishore Dey, Sivaramakrishnan Sivasubramanian
Rok vydání: 2020
Předmět:
Zdroj: Annals of Combinatorics. 24:711-738
ISSN: 0219-3094
0218-0006
DOI: 10.1007/s00026-020-00511-6
Popis: The Eulerian polynomial $$ \mathrm {AExc}_n(t)$$ enumerating excedances in the symmetric group $$\mathfrak {S}_n$$ is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider $$ \mathrm {AExc}_n^+(t)$$ and $$ \mathrm {AExc}_n^-(t)$$ , the polynomials which enumerate excedance in the alternating group $$\mathcal {A}_n$$ and in $$\mathfrak {S}_n - \mathcal {A}_n$$ , respectively. We show that $$ \mathrm {AExc}_n^+(t)$$ is gamma positive iff $$n \ge 5$$ is odd. When $$n \ge 4$$ is even, $$ \mathrm {AExc}_n^+(t)$$ is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about $$ \mathrm {AExc}_n^-(t)$$ . We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in $$\mathfrak {S}_n$$ . We extend some of these to the case when enumeration is done over even and odd derangements in $$\mathfrak {S}_n$$ .
Databáze: OpenAIRE
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