The Eulerian distribution on involutions is indeed unimodal

摘要

Let I-n,I-k (respectively J(n,k)) be the number of involutions (respectively fixed-point free involutions) of {1,...,n) with k descents. Motivated by Brenti's conjecture which states that the sequence I-n,(0),I-n,(1),...,I-n,(n-1) is log-concave, we prove that the two sequences I-n,I-k and J(2n,k) are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a(n,k) such that