Combinatorial results for semigroups of order-preserving partial transformations

Let be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that satisfies the recurrence with initial conditions c0=1, c1=2. It is also shown that satisfies the recurrence en+1=5(en−en−1)+1 with initial conditions e0=1, e1=2. Moreover, the cardinalities of the Greenʹs relations , and have been computed.