Lattice paths and order-preserving partial transformations

Abstract

Let PO_n be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that there exist bijections between the set of certain lattice paths in the Cartesian plane that start at (0,0), end at (n - 1,n - 1), and.certain subsemigroups of PO_n. Several consequences of these bijections were discussed.