The Eulerian distribution on involutions is indeed γ-positive

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:165

Article

Wang, Danielle¹ 

[1] MIT, Dept Math, Cambridge, MA 02139 USA

关键词: Involutions;    Descent number;    gamma-Positive;    Eulerian polynomial;    Separable permutations;   

DOI  :  10.1016/j.jcta.2019.02.004

来源: Elsevier

【 摘 要 】

Let I-n and J(n) denote the set of involutions and fixed-point free involutions of {1, ... , n}, respectively, and let des(pi) denote the number of descents of the permutation pi. We prove a conjecture of Guo and Zeng which states that I-n(t) := Sigma(pi is an element of In) t(des(pi)) is gamma-positive for n >= 1 and J(2n) (t) := Sigma(pi is an element of J2n) t(des(pi)) is gamma-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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free