On fixed points of permutations

2007/08/20 by Persi Diaconis, Jason Fulman, Diaconis, Persi +3 · 3 citations
Computer Science · Mathematics · #05A16 #20B30 #20B35 #60C07 #Advanced Algebra and Geometry #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.0708.2643

openalex publication_date 2007/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Rough permutations with a fixed set of a given size
• Derangements in affine classical groups and Cohen-Lenstra heuristics
• Number of orbits of k-subsets of permutations

Related

• Fixed points of non-uniform permutations and representation theory of the symmetric group
• Unseparated pairs and fixed points in random permutations
• Fixed points, descents, and inversions in parabolic double cosets of the symmetric group
• Limit theorems for fixed point biased permutations avoiding a pattern of length three
• On statistics of permutations chosen from the Ewens distribution