Poisson approximations of the number of fixed points in random multiset permutations

For ordinary permutations on $n$ letters, the distribution of the number of fixed points of random permutations is well known to approach the Poisson$(1)$ distribution in total variation distance as $n\to \mathrm{\infty }$ super-exponentially quickly. We use Stein’s method to get related results for the number of fixed points of random permutations of multisets. Given a sequence of multisets on $n$ letters whose expected number of fixed points converges to a constant $c$, we must have $c\ge 1$ and the distribution of number of fixed points converges to the $\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}(c)$ distribution as $n\to \mathrm{\infty }$. If $c>1$, then the rate of convergence in total variation distance can be as slow as $n^{-1\mathrm{/}2}$.