Unseparated pairs and fixed points in random permutations

TL;DR

This paper explores the distribution of certain element sets in random permutations, providing multiple proofs and extending results to circular permutations, with implications for understanding fixed points and related structures.

Contribution

It offers three different proofs of a key distributional fact in permutations and extends the analysis to circular permutations and commutators, with bounds on distributional approximations.

Findings

The set of elements where \Pi(k+1) = \\Pi(k) + 1 has the same distribution as fixed points in [n-1].

Distribution of analogous sets in circular permutations is characterized.

Total variation distance between fixed points of commutators and Poisson distribution is small for large n.

Abstract

In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this fact using, respectively, an enumeration relying on the inclusion-exclusion principle, the introduction of two different Markov chains to generate uniform random permutations, and the construction of a combinatorial bijection. We also obtain the distribution of the analogous set for circular permutations that consists of those k in [n] such that \Pi(k+1 mod n) = \Pi(k) + 1 mod n. This latter random set is just the set of fixed points of the commutator [\rho, \Pi], where \rho is the n-cycle (1,2,...,n). We show for a general permutation \eta that, under weak conditions on the number of fixed points and 2-cycles of \eta, the total variation distance between the distribution of the number of fixed points of [\eta,\Pi] and a Poisson distribution with expected value 1 is small when n is large.