The Eulerian Distribution on Involutions is Indeed Unimodal

Abstract

Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. 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,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 $$\sum _{k=0}^{n-1}{I}_{n,k}{t}^k=\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{a}_{n,k}{t}^k(1+t{)}^{n-2k-1}.$$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right.

12 pages, minor changes, to appear in J. Combin. Theory Ser. A

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• Plethystic formulas for permutation enumeration
• The Eulerian distribution on self evacuated involutions