On the ranks of certain subsemigroups of finite orientation-preserving and order-decreasing partial transformations

Let POPDn be the semigroup consisting of all orientation-preserving and order-decreasing partial transformations on the finite chain X-n = {1 < < n}, and let POPD(n,r)={ alpha is an element of POPDn:|im(alpha)| <= r} for 1 <= r <= n-1. We prove that the rank and the idempotent rank of POPD (n,r) are both equal to (n)& sum;(s=r)(ns)(sr)+(2n-r-1)(r-2)/2 for 1 <= r <= n-1. Then we conclude that the rank and the idempotent rank of POPDn are both equal to n(2)+n+2/2.