The Eulerian distribution on involutions is indeed γ-positive

2018/08/25 by Danielle Wang, Wang, Danielle · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1808.08481

openalex publication_date 2018/08/25 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Let \mathcal I_n and \mathcal J_n denote the set of involutions and fixed-point free involutions of \1, …, n\, respectively, and let des(π) denote the number of descents of the permutation π. We prove a conjecture of Guo and Zeng which states that I_n(t) := ∑_{π∈ \mathcal In} t^des(π) is γ-positive for n ≥ 1 and J_2n(t) := ∑_π∈ \mathcal J_2n t^des(π) is γ-positive for n ≥ 9. We also prove that the number of (3412, 3421)-avoiding permutations with m double descents and k descents is equal to the number of separable permutations with m double descents and k descents.

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