Combinatorial results for certain semigroups of order-decreasing partial isometries of a finite chain

Let ${\mathcal{I}}_n$ be the symmetric inverse semigroup on $X_n = {1, 2, \ldots , n}$andlet${\cal DDP}_n$and${\cal ODDP}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving and order-decreasing partial isometries of $X_n$, respectively. In this paper we investigate the cardinalities of some equivalences on ${\mathcal{D}\mathcal{D}\mathcal{P}}_n$ and ${\cal ODDP}_n$ which lead naturally to obtaining the order of the semigroups