Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets

2017/01/17 by Pinsky, Ross G.
#05A05 #05A15 #60C05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1701.04855

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 m, is the number of sets of size m 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.

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