On fixed points of permutations

TL;DR

This paper investigates the limiting distributions of fixed points in various group actions, extending classical results on permutations and providing a complete classification for primitive actions.

Contribution

It offers a comprehensive classification of limiting fixed point distributions for primitive group actions, including new asymptotic estimates and a survey of related findings.

Findings

Most primitive actions have trivial fixed point distributions

For the symmetric group's action on k-sets, the limit involves polynomials in Poisson variables

The classification exhausts all primitive actions with non-trivial fixed point limits

Abstract

The number of fixed points of a random permutation of 1,2,...,n has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial -- almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of 1,2,...,n, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results.