Asymptotic normality of pattern counts in conjugacy classes

TL;DR

This paper proves that pattern counts in permutations from conjugacy classes are asymptotically normally distributed, with non-degenerate variance, using weighted dependency graphs.

Contribution

It establishes the asymptotic normality of vincular pattern counts in conjugacy classes under mild conditions and confirms the non-degeneracy of the limiting variance.

Findings

Pattern counts are asymptotically normal in conjugacy classes.

Limiting variance is always non-degenerate for classical patterns.

Weighted dependency graphs are used in the proof.

Abstract

We prove, under mild conditions on fixed points and two cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class.Additionally, we prove that the limiting variance is always non-degenerate for classical pattern counts. The proof uses weighted dependency graphs.