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arXiv · 2602.11061

A statistic-swapping involution on the Cartesian product of the symmetric group $S_{kn}$ and the generalized symmetric group $S(k,n)$

Peter Kagey·Kai Mawhinney
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Abstract

We construct a statistic-swapping involution on the Cartesian product of the generalized symmetric group $S(k,n)$ with the symmetric group $S_{kn}$, which swaps the number of fixed points in the generalized symmetric group element with the number of $k$-cycles in the symmetric group element. This gives a combinatorial proof for a probabilistic observation: the distribution of fixed points on $S(k,n)$ matches the distribution of $k$-cycles on $S_{kn}$.

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Peter Kagey, Kai Mawhinney. 2026-02-11. A statistic-swapping involution on the Cartesian product of the symmetric group $S_{kn}$ and the generalized symmetric group $S(k,n)$. https://arxiv.org/abs/2602.11061

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