  Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots
, n\}$ and let ${\cal DDP}_n$ 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 ${\cal DDP}_n$ and ${\cal
ODDP}_n$ which lead naturally to obtaining the order of the semigroups
