The Eulerian distribution on involutions is indeed γ-positive

Let I n and 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∈I n t ^des(π) is γ-positive for n≥1 and J 2n (t):=∑ π∈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.

Duke Scholars

Author Danielle Y Wang Mathematics