Geodesic
Averaging algebras, Schröder numbers, rooted trees and operads
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 73-109.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 Schröder numbers, leading us to give interpretations of large Schröder 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.
Classification : 05E15, 16S10, 16W99, 18D50
Keywords: averaging operator, averaging algebra, free object, bracketed words, large Schröder numbers, generating function, rooted trees, operad 
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     title = {Averaging algebras, {Schr\"oder} numbers, rooted trees and operads},
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Pei, Jun; Guo, Li. Averaging algebras, Schröder numbers, rooted trees and operads. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 73-109. http://geodesic.mathdoc.fr/item/JAC_2015__42_1_a8/