Asymptotic results for semigroups of order-preserving partial transformations

Let script P sign script C sign n be the semigroup of all decreasing and order-preserving partial transformations of an n -element chain, and let E (script P sign script C sign n ) be its set of idempotents. Among other results, asymptotic formulae for |script P sign script C sign n | and | E (script P sign script C sign n )|/|script P sign script C sign n | are obtained. Similar results for script P sign script O sign n the (larger) semigroup of all order-preserving partial transformations of an n -element chain are also obtained.