On asymptotically free action of permutation groups on subsets and multisets

Abstract

Let $G$ be a permutation group acting on a finite set $Ω$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $Ω_m$ of all size $m$ subsets of $Ω$ satisfies the trivial inequalities $|Ω_m|/|G|\leq |Ω_m/G|\leq |Ω_m|$. The paper offers improvements of the upper bound in terms of the minimal degree of $G$ or the minimal degree of some its subset with a small complement. Applications include asymptotic enumeration of point configurations in an affine space over a finite field, unlabeled graphs and hypergraphs. Finally, with references to known results of permutation groups theory it is shown that if $G$ is an arbitrary 2-transitive group except for $S_n$ and $A_n$, then $|Ω_m/G|\approx |Ω_m|/|G|$ for $m$ and $n$ large provided the ratio $m/n$ is bounded away from 0 and 1. Similar results hold for the induced action of $G$ on the set $Ω_{(m)}$ of all weight $m$ multisets on $Ω$ provided the ratio $m/n$ is not too small.