## Electronic Communications in Probability

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

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

#### 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**

Received: 30 January 2017

Accepted: 14 February 2017

First available in Project Euclid: 18 February 2017

**Permanent link to this document**

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

**Keywords**

permutations that fix a set covers of multisets Touchard polynomials Dobínski’s formula Bell numbers cycles in random permutations Ewens sampling distribution

**Rights**

Creative Commons Attribution 4.0 International License.

#### 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