The average number of stable matchings
SIAM Journal on Discrete Mathematics
Regular Article: On a Likely Shape of the Random Ferrers Diagram
Advances in Applied Mathematics
Combinatorics of Permutations
Transitivity And Connectivity Of Permutations
Combinatorica
Analytic Combinatorics
Indecomposable permutations, hypermaps and labeled Dyck paths
Journal of Combinatorial Theory Series A
| Hi-index | 0.00 |
A permutation @s of [n] induces a graph G"@s on [n] - its edges are inversion pairs in @s. The graph G"@s is connected if and only if @s is indecomposable. Let @s(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(@s(n,m) is indecomposable) and P(G"@s"("n","m") is connected). We prove that p(n,m) is non-decreasing with m by constructing a Markov process {@s(n,m)} in which @s(n,m+1) is obtained by increasing one of the components of the inversion sequence of @s(n,m) by one. We show that, with probability approaching 1, G"@s"("n","m") becomes connected for m asymptotic to m"n=(6/@p^2)nlogn. We also find the asymptotic sizes of the largest and smallest components when the number of edges is moderately below the threshold m"n.