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

Ross G. Pinsky

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arXiv · 1701.04855

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

Ross G. Pinsky

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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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BibTeX RIS

Ross G. Pinsky. 2017-01-20. Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets. https://arxiv.org/abs/1701.04855

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