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

A permutation σ of [ n ] induces a graph G σ on [ n ] – its edges are inversion pairs in σ. The graph G σ 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 ( G σ ( 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, G σ ( n , m ) becomes connected for m asymptotic to m n = ( 6 / π 2 ) n log n . We also find the asymptotic sizes of the largest and smallest components when the number of edges is moderately below the threshold m n .