Combinatorial results for semigroups of orientation-preserving partial transformations

Let X_n = {1, 2,..., n}. On a partial transformation α: Dom α ⊆ Xn! Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is {pipe} Dom α {pipe}, the height of α is {pipe} Im α {pipe}, and the right (resp., left) waist of α is max(Im α) (resp., min(Im α)). We compute the cardinalities of some equivalences defined by equalities of these parameters on OP_n, the semigroup of orientation-preserving full transformations of X_n, POP_n the semigroup of orientation-preserving partial transformations of X_n, OR_n the semigroup of orientation-preserving/reversing full transformations of X_n, and POR_n the semigroup of orientation-preserving/reversing partial transformations of X_n, and their partial one-to-one analogue semigroups, POPI_n and PORI_n.