Wilf classes of non-symmetric operads

TL;DR

This paper explores the classification of non-symmetric operads into Wilf classes based on their generating series, introduces algorithmic approaches for operads with finite Groebner bases, and refines existing results on binary operads with a single relation.

Contribution

It provides a new perspective on classifying operads via Wilf classes, establishes the connection with formal language theory, and improves calculations related to binary operads with one relation.

Findings

Operads with finite Groebner bases have monomial bases forming unambiguous context-free languages.

An algorithmic method to compute generating series for certain operads is proposed.

Empirical confirmation of Rowland's conjecture on pattern avoidance in binary trees.

Abstract

Two operads are said to belong to the same Wilf class if they have the same generating series. We discuss possible Wilf classifications of non-symmetric operads with monomial relations. As a corollary, this would give the same classification for the operads with a finite Groebner basis. Generally, there is no algorithm to decide whether two finitely presented operads belong to the same Wilf class. Still, we show that if an operad has a finite Groebner basis, then the monomial basis of the operad forms an unambiguous context-free language. Moreover, we discuss the deterministic grammar which defines the language. The generating series of the operad can be obtained as a result of an algorithmic elimination of variables from the algebraic system of equations defined by the Chomsky--Schutzenberger enumeration theorem. We then focus on the case of binary operads with a single relation. The approach is based on the results by Rowland on pattern avoidance in binary trees. We improve and refine Rowland's calculations and empirically confirm his conjecture. Here we use both the algebraic elimination and the direct calculation of formal power series from algebraic systems of equations. Finally, we discuss the connection of Wilf classes with algorithms for the Quillen homology of operads calculation.