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Combinatorial results for order-preserving partial injective contraction mappings

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.