m-Potent Elements in Order-Preserving Transformation Semigroups and Ordered Trees
Orhan Sönmez, Yusuf Ünlü · Communications in Algebra · 2013
Let 𝒞 n, r , 𝒲 n, r , and 𝒰 n, r be the semigroups of s-potent elements α in 𝒪 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 |𝒞 n, r |, |𝒲 n, r |, and |𝒰 n, r |. We construct a correspondence between 𝒞 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 n, r |.