The descent statistic on involutions is not log-concave

TL;DR

This paper links involution descent counts to Young tableaux enumeration, demonstrating that the sequence is not log-concave for some cases, thus resolving a conjecture by Brenti.

Contribution

It establishes a new combinatorial connection between involutions with descents and Young tableaux, and disproves the log-concavity conjecture.

Findings

Sequences are not log-concave for some n

Established a combinatorial link between involutions and Young tableaux

Resolved a conjecture by F. Brenti

Abstract

We establish a combinatorial connection between the sequence $(i_{n,k})$ counting the involutions on $n$ letters with $k$ descents and the sequence $(a_{n,k})$ enumerating the semistandard Young tableaux on $n$ cells with $k$ symbols. This allows us to show that the sequences $(i_{n,k})$ are not log-concave for some values of $n$, hence answering a conjecture due to F. Brenti.