  Let $ \mathcal{I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2,
\ldots , n\}$. Let $\mathcal{OCI}_n$ be the subsemigroup of $\mathcal{I}_n$
consisting of all order-preserving injective partial contraction mappings, and
let $\mathcal{ODCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all
order-preserving and order-decreasing injective partial contraction mappings of
$X_n$. In this paper, we investigate the cardinalities of some equivalences on
$\mathcal{OCI}_n$ and $\mathcal{ODCI}_n$ which lead naturally to obtaining the
order of these semigroups. Then, we relate the formulae obtained to Fibonacci
numbers. Similar results about $\mathcal{ORCI}_n$, the semigroup of
order-preserving or order-reversing injective partial contraction mappings, are
deduced.
