Title: Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials 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.