Some counting problems related to permutation groups

14 years 8 days ago

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This paper discusses investigations of sequences of natural numbers which count the orbits of an in nite 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 are su ciently special to be interesting but su ciently common to support a general theory. `I count a lot of things that there's no need to count,' Cameron said. `Just because that's the way I am. But I count all the things that need to be counted.' Richard Brautigan, The Hawkline Monster 1 Three counting problems This paper is a survey of the problem of counting the orbits of an in nite permutationgroup on n-sets or n-tuples, especiallythe aspects closest to algebraic combinatorics. Much of the material surveyed here can be found elsewhere, for example in 4]. We begin by dis...

Peter J. Cameron

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