A statistic-swapping involution on the Cartesian product of the symmetric group $S_{kn}$ and the generalized symmetric group $S(k,n)$
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 $S_{kn}$.
Fields of Study & Subjects
Advanced Combinatorial Mathematics
Finite Group Theory Research
Probability and Statistical Research
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 11 February 2026
- Language
- EN
- Document type
- Preprint