Orbits on n-tuples

Probability of Mutually Commuting N-Tuples in Some Classes of Compact Groups

Some counting problems related topermutation

This paper discusses investigations of sequences of natural numbers which count the orbits of an innnite permutation group on n-sets or n-tuples. It surveys known results on the growth rates, cycle index techniques, and an interpretation as the Hilbert series of a graded algebra, with a possible application to the question of smoothness of growth. I suggest that these orbit-counting sequences a...

Degeneration and orbits of tuples and subgroups in abelian groups

A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm’s...

Groups and sequences

The purpose of this paper is to identify, as far as possible, those sequences in the Encyclopedia of Integer Sequences which count orbits of an infinite permutation group acting on n-sets or n-tuples of elements of the permutation domain. The paper also provides an introduction to the properties of such sequences and their relations with combinatorial enumeration problems.

Sharply $(n-2)$-transitive Sets of Permutations

Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called e...