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

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.