Cross work induced by transformations on the symmetric group and related transformation semigroups

We introduce a new permutation statistic, called the \emph{cross work}, on four important subsemigroups of the partial transformation semigroup $\mathcal{PT}_n$: the symmetric group $\mathcal{S}_n$, the semigroup of singular transformations $\mathrm{Sing}_n$, the symmetric inverse semigroup $\mathcal{I}_n$, and the semigroup $\mathcal{POI}_n$ 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 $\mathcal{S}_n$. We also determine the exact minimum and maximum values of the cross work, showing that the maximal value grows quadratically with $n$.