COMBINATORIAL RESULTS FOR ORDER-PRESERVING PARTIAL INJECTIVE CONTRACTION MAPPINGS

Let I-n be the symmetric inverse semigroup on X-n = { 1 , 2 , ... , n } . Let OCIn be the subsemigroup of I n consisting of all order -preserving injective partial contraction mappings, and let ODCIn be the subsemigroup of 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 OCIn and ODCIn which lead naturally to obtaining the order of these semigroups. Then, we relate the formulae obtained to Fibonacci numbers. Similar results about ORCIn , the semigroup of order -preserving or orderreversing injective partial contraction mappings, are deduced.