Combinatorial results for certain semigroups of partial isometries of a finite chain

Let In be the symmetric inverse semigroup on X_n = {1,2,..., n} under composition of maps and let DP_n and ODP_n be its subsemigroups of partial isometries and of order-preserving partial isometries of X_n, respectively. In this paper we investigate the cardinalities of some equivalences on DP_n and ODP_n which lead naturally to obtaining the orders of these semigroups.