Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets
Probability
2017-01-23 v2
Abstract
We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by , is the number of sets of size fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.
Keywords
Cite
@article{arxiv.1701.04855,
title = {Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets},
author = {Ross G. Pinsky},
journal= {arXiv preprint arXiv:1701.04855},
year = {2017}
}
Comments
In this version, Theorem C is stated and proved for the Ewens sampling distributions, not just for the uniform distribution. Also, a few typos have been corrected