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

Ross G Pinsky

doi:10.1214/17-ecp49

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.

Highlights

Introduction and Statement of ResultsIn this paper, we present a new and perhaps simpler 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 present a new and perhaps simpler 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 introduce a rather novel permutation statistic and study its distribution

Summary

Introduction and Statement of Results

We present a new and perhaps simpler 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 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. Permutations that fix a set, covers of multisets, Touchard Polynomials, Dobınski’s formula, Bell numbers, cycles in random permutations, Ewens sampling distribution. The Bell number Bn denotes the number of partitions of a set of n distinct elements. Using (1.8) and the fact that (k)j = 0 for j > k, we can write the nth moment μn;λ of a Pois(λ)-distributed random variable as (1.9). Under Pn,θ, the random vector (C1(n), C2(n), · · · , Cm(n), . . .) converges weakly to (Zθ, Z θ , · · · , Z θ , · · · ), where the random variables {Z θ }∞ m=1 m m are independent, and

Zθ m θ m

Xnk exists for all k

1Dj if and only if for some l

Given the