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.
Citation
Ross G. Pinsky. "Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets." Electron. Commun. Probab. 22 1 - 9, 2017. https://doi.org/10.1214/17-ECP49
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