Averaging algebras, Schroder numbers, rooted trees and operads

In this paper, we study averaging operators from an algebraic and combinatorial point of view. We first construct free averaging algebras in terms of a class of bracketed words called averaging words. We next apply this construction to obtain generating functions in one or two variables for subsets of averaging words when the averaging operator is taken to be idempotent. When the averaging algebra has an idempotent generator, the generating function in one variable is twice the generating function for large Schroder numbers, leading us to give interpretations of large Schroder numbers in terms of bracketed words and rooted trees, as well as a recursive formula for these numbers. We also give a representation of free averaging algebras by unreduced trees and apply it to give a combinatorial description of the operad of averaging algebras.