On the connected components of a random permutation graph with a given number of edges

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Abstract

A permutation σ of [n] induces a graph Gσ on [n] – its edges are inversion pairs in σ. The graph Gσ is connected if and only if σ is indecomposable. Let σ(n,m) denote a permutation chosen uniformly at random among all permutations of [n] with m inversions. Let p(n,m) be the common value for the probabilities P(σ(n,m) is indecomposable) and P(Gσ(n,m) is connected). We prove that p(n,m) is non-decreasing with m by constructing a Markov process {σ(n,m)} in which σ(n,m+1) is obtained by increasing one of the components of the inversion sequence of σ(n,m) by one. We show that, with probability approaching 1, Gσ(n,m) becomes connected for m asymptotic to mn=(6/π2)nlogn. We also find the asymptotic sizes of the largest and smallest components when the number of edges is moderately below the threshold mn.

Keywords

Random permutation
Permutation graph
Connectivity threshold
Indecomposable permutation
Inversion

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Supported in part by NSF Grant DMS-1101237.