Cross work induced by transformations on the symmetric group and related transformation semigroups
Abstract
We introduce a new permutation statistic, called the \emph{cross work}, on four important subsemigroups of the partial transformation semigroup 𝒫𝒯n: the symmetric group 𝒮n, the semigroup of singular transformations Singn, the symmetric inverse semigroup ℐn, and the semigroup 𝒫𝒪ℐn of partial order-preserving injective transformations. The cross work is defined via local distance interactions between adjacent positions and their images. Fundamental invariance properties are established, including inversion invariance under a natural symmetry condition and a characterization of conjugations preserving the statistic. Using elementary combinatorial methods, we derive closed formulas for the total and average cross work on 𝒮n. We also determine the exact minimum and maximum values of the cross work, showing that the maximal value grows quadratically with n.
Related
- Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups
- Maximal regular subsemigroups containing the group of units of transformation semigroups that preserve a uniform equivalence relation with two classes
- Quantum Cheeger Inequalities for KMS-Symmetric Quantum Markov Semigroups
- Some entropic inequalities for primitive KMS symmetric quantum Markov semigroups
- The Differential Structure of Generators of KMS-Symmetric Quantum Markov Semigroups
BibTeX
@misc{udo2026cross,
title = {{Cross work induced by transformations on the symmetric group and related transformation semigroups}},
author = {Nsibiet E. Udo and Victor A. Owai and Moses O. Francis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34902},
eprint = {2610.34902},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34902},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cross work induced by transformations on the symmetric group and related transformation semigroups
AU - Nsibiet E. Udo
AU - Victor A. Owai
AU - Moses O. Francis
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34902
UR - https://arxiv.org/abs/2610.34902
AB - We introduce a new permutation statistic, called the \emph{cross work}, on four important subsemigroups of the partial transformation semigroup 𝒫𝒯_n: the symmetric group 𝒮_n, the semigroup of singular transformations Sing_n, the symmetric inverse semigroup ℐ_n, and the semigroup 𝒫𝒪ℐ_n of partial order-preserving injective transformations. The cross work is defined via local distance interactions between adjacent positions and their images. Fundamental invariance properties are established, including inversion invariance under a natural symmetry condition and a characterization of conjugations preserving the statistic. Using elementary combinatorial methods, we derive closed formulas for the total and average cross work on 𝒮_n. We also determine the exact minimum and maximum values of the cross work, showing that the maximal value grows quadratically with n.
ER -