Asymptotically free action of permutation groups on subsets and multisets

Abstract

Let be a permutation group acting on a nite set of cardinality . The number of orbits of the induced action of on the set of all -element subsets of obeys the trivial estimates | |/| | ≤ | / | ≤ | |. In this paper the upper estimate is improved in terms of the minimal degree of the group or the minimal degree of its subset with small complement. In particular, using the universal estimates obtained by Bochert for the minimal degree of a group and by Babai-Pyber for the order of a group, in terms of only we demonstrate that if is a 2-transitive group other than the full symmetric or the alternating groups, and are large enough, and the ratio / is bounded away from 0 and 1, then | / | ≈ | |/| |. Similar results hold true for the induced action of on the set ( ) of all -element multisets with elements drawn from , provided that the ratio /( + ) is uniformly bounded away from 0 and 1.Keywords: permutation group, regular orbits, average size of the stabilizer, minimal degree of a group, asymptotics of the number of orbits, enumeration of a ne con gurations, enumeration of graphs, asymptotically free action.