m-POTENT ELEMENTS IN ORDER-PRESERVING TRANSFORMATION SEMIGROUPS AND ORDERED TREES

摘要

Let e_(n,r), W_(n,r), and U_(n,r) be the semigroups of s-potent elements α in e_n, the semigroup of order-preserving mappings, such that Fix(α) = {1}, |Fix(α)| = 1, and Fix(α) is arbitrary, respectively. We give combinatorial results relating the cardinalities |e_(n,r)|, |W_(n,r), and |U_(n,r)|. We construct a correspondence between e_(n,r) and T_(n,r) the ordered trees with depth at most r and n nodes. Using this we produce generating functions for |W_(n,r)|,|U_9n,r)|.