The Eulerian distribution on involutions is indeed gamma-positive

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.