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 $\mathca