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

TL;DR

The paper introduces a combinatorial involution that swaps fixed points and k-cycles between the generalized symmetric group and the symmetric group, providing a new proof for their distributional equivalence.

Contribution

It constructs a novel statistic-swapping involution on the product of two groups, linking fixed points and k-cycles in a combinatorial framework.

Findings

Establishes a bijective involution swapping fixed points and k-cycles.

Provides a combinatorial proof for the distributional equivalence.

Connects properties of generalized symmetric groups with classical symmetric groups.

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 $S_{kn}$.