Longest increasing subsequences and log concavity

TL;DR

This paper explores the log concavity of the distribution of longest increasing subsequences in permutations and involutions, relating these properties to the Robinson-Schensted algorithm and limiting distributions.

Contribution

It introduces conjectures on log concavity for permutations and involutions, and connects these to shape distributions via the Robinson-Schensted algorithm, providing new insights and partial proofs.

Findings

Conjecture that the sequence of counts of permutations by LIS length is log concave.

Establishes a close relationship between permutation and involution cases.

Provides a proof of log concavity for part of the limiting distribution.

Abstract

Let $\pi$ be a permutation of $[n]=\{1,\dots,n\}$ and denote by $\ell(\pi)$ the length of a longest increasing subsequence of $\pi$. Let $\ell_{n,k}$ be the number of permutations $\pi$ of $[n]$ with $\ell(\pi)=k$. Chen conjectured that the sequence $\ell_{n,1},\ell_{n,2},\dots,\ell_{n,n}$ is log concave for every fixed positive integer $n$. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Schensted algorithm have certain shapes. In addition, we present a proof of Deift that part of the limiting distribution is log concave. Various other conjectures are discussed.