Groups of permutations preserving orientation (parity) of subsets of a fixed size, and related monoids
Vitor Fernandes, Alexei Vernitski
TL;DR
The paper addresses the problem of classifying permutation groups on $[n]$ that preserve the parity (inversion parity) on all $t$-element subsets, introducing the groups $\Gamma_n^t$ and corresponding monoids $\Sigma_n^t$ that generalize classical order-preserving and orientation-preserving structures. It provides a complete $t$-dependent classification: $\Gamma_n^2=\{\iota_n\}$, $\Gamma_n^3=Z_n$, and $\Gamma_n^n=A_n$, with $2\le t\le n-2$ yielding a four-way split (trivial, cyclic, dihedral) dictated by $t\pmod4$, and shows $\Gamma_n^{n-1}$ forms half of the parity-alternating permutations PAP$_n$. The work then develops monoids $\Sigma_n^t$ via width-$t$ injective restrictions, establishes their relations to classical monoids (e.g., $O_n$, $OP_n$, $M_n$, $OR_n$), and uncovers periodic embedding patterns among these monoids through $\Delta_n^t$, whose units recover $\Gamma_n^t$; several nontrivial inclusions and counterexamples illustrate the structure. These results connect parity-based permutation groups with well-known symmetry groups and monoid theory, offering width-based criteria for monotone and oriented mappings and opening avenues for further generalizations of monoid families.
Abstract
We study permutations on n elements preserving orientation (parity) of every subset of size k. We describe all groups of these permutations. Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral. In this context, we define and study monoids generalizing monoids of order-preserving mappings and monoids of orientation-preserving mappings.