Enumeration of certain finite semigroups of transformations

Let Singn be the semigroup of singular self-maps of Xn = {1, … ,n}, let Rn = {α ∈ Singn: (∀y ∈ Im α)¦yα−1¦⩾¦Im α¦} and let E(Rn) be the set of idempotents of Rn. Then it is shown that Rn = (E(Rn))2. Moreover, expressions for the order of Rn and E(Rn) are obtained in terms of the kth-upper Stirling number of the second kind, S(n, r, k); defined as the number of partitions of Xn into r subsets each of size not less than k.