# Combinatorial results for order-preserving partial injective contraction mappings

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

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Georg Klein