Asymptotically free action of permutation groups on subsets and multisets
-
Sergey Yu. Sadov
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 Ωm of all m-element subsets of Ω obeys the trivial estimates |Ωm|/|G| ≤ |Ωm/G| ≤ |Ωm|. In this paper the upper estimate is improved in terms of the minimal degree of the group G 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 n only we demonstrate that if G is a 2-transitive group other than the full symmetric or the alternating groups,mand n are large enough, and the ratio m/n is bounded away from 0 and 1, then |Ωm/G| ≈ |Ωm|/|G|. Similar results hold true for the induced action of G on the set Ω(m) of all m-element multisets with elements drawn from Ω, provided that the ratio m/(m + n) is uniformly bounded away from 0 and 1.
© 2015 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Nonnegative basis of a lattice
- Application of non-associative groupoids to the realization of an open key distribution procedure
- An approach to the classification of Boolean bent functions of the nonlinearity degree 3
- Asymptotically free action of permutation groups on subsets and multisets
- The structure of finite abelian n-ary groups
- Analysis of a discrete semi-Markov model of continuous inventory control with periodic interruptions of consumption
Articles in the same Issue
- Frontmatter
- Nonnegative basis of a lattice
- Application of non-associative groupoids to the realization of an open key distribution procedure
- An approach to the classification of Boolean bent functions of the nonlinearity degree 3
- Asymptotically free action of permutation groups on subsets and multisets
- The structure of finite abelian n-ary groups
- Analysis of a discrete semi-Markov model of continuous inventory control with periodic interruptions of consumption
