Log-concavity, the Ulam distance and involutions

TL;DR

This paper demonstrates that for many natural sets of permutations, the count of permutations at a fixed Ulam distance from the identity exhibits log-concavity, revealing a new structural property of permutation sets.

Contribution

It establishes the log-concavity of permutation counts at fixed Ulam distances in a broad class of permutation sets, a novel insight into permutation combinatorics.

Findings

Permutation counts at fixed Ulam distance are log-concave.

Log-concavity holds for a large collection of permutation sets.

Provides new structural understanding of permutation distributions.

Abstract

We prove that in a large collection of naturally defined sets of permutations of fixed length, the numbers of permutations at Ulam distance k from the identity form a log-concave sequence in k.