Fixed points of non-uniform permutations and representation theory of the symmetric group

TL;DR

This paper employs symmetric group representation theory to establish Poisson limit theorems for fixed points in various non-uniform permutation models, extending understanding of permutation structure and distributional limits.

Contribution

It introduces new Poisson limit theorems for fixed points in non-uniform permutations using representation theory, covering commutators and specific cycle structures.

Findings

Poisson limit for commutators of uniform permutations

Poisson limit for fixed permutations with one uniform component

Results for permutations with many random cycles

Abstract

We use representation theory of the symmetric group S_n to prove Poisson limit theorems for the distribution of fixed points for three types of non-uniform permutations. First, we give results for the commutator of g and x where g and x are uniform in S_n. Second, we give results for the commutator of g and x where g in uniform in S_n and x is fixed. Third, we give results for permutations obtained by multiplying n*log(n)/i + cn many random i-cycles. Some of our results are known by other, quite different, methods.