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

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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.

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


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