Combinatorial results for semigroups of order-preserving partial transformations

Let POn be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that \POn\ = c(n) satisfies the recurrence (2n - 1)(n + 1)c(n+1) = 4(3n(2) - 1)c(n) - (2n + 1)(n - 1)c(n-1) with initial conditions c(0) = 1, c(1) = 2. It is also shown that \E(POn)\ = e(n) satisfies the recurrence e(n+1) = 5(e(n) - e(n-1)) + 1 with initial conditions e(0) = 1, e(1) = 2. Moreover, the cardinalities of the Green's relations L, R and J have been computed. (C) 2003 Elsevier Inc. All rights reserved.