## Electronic Communications in Probability

### Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets

Ross G. Pinsky

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 17, 9 pp.

Dates
Accepted: 14 February 2017
First available in Project Euclid: 18 February 2017

https://projecteuclid.org/euclid.ecp/1487386905

Digital Object Identifier
doi:10.1214/17-ECP49

Mathematical Reviews number (MathSciNet)
MR3615668

Zentralblatt MATH identifier
06691236

Subjects
Primary: 60C05, 05A05, 05A15

#### Citation

Pinsky, Ross G. Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets. Electron. Commun. Probab. 22 (2017), paper no. 17, 9 pp. doi:10.1214/17-ECP49. https://projecteuclid.org/euclid.ecp/1487386905

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