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- W2078369576 abstract "Let $G$ be a permutation group acting on a finite set $Omega$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $Omega_m$ of all size $m$ subsets of $Omega$ satisfies the trivial inequalities $|Omega_m|/|G|leq |Omega_m/G|leq |Omega_m|$. 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 $|Omega_m/G|approx |Omega_m|/|G|$ 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 $Omega_{(m)}$ of all weight $m$ multisets on $Omega$ provided the ratio $m/n$ is not too small." @default.
- W2078369576 created "2016-06-24" @default.
- W2078369576 creator A5059128219 @default.
- W2078369576 date "2013-12-24" @default.
- W2078369576 modified "2023-09-27" @default.
- W2078369576 title "On asymptotically free action of permutation groups on subsets and multisets" @default.
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- W2078369576 doi "https://doi.org/10.1515/dma-2015-0004," @default.
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