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arXiv · 2608.00710

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

Dudley Stark
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Abstract

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 $n\to\infty$ 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 $c\geq 1$ and the distribution of number of fixed points converges to the ${\rm Poisson}(c)$ distribution as $n\to\infty$. If $c>1$, then the rate of convergence in total variation distance can be as slow as $n^{-1/2}$.

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Dudley Stark. 2026-08-01. Poisson approximations of the number of fixed points in random multiset permutations. https://arxiv.org/abs/2608.00710

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