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.

中文翻译：

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非对称操作数的Wilf类

如果两个操作员具有相同的生成序列，则它们被称为属于同一Wilf类。我们讨论具有单项式关系的非对称操作者的可能Wilf分类。作为必然结果，这将为具有有限Groebner基础的操作员提供相同的分类。通常，没有算法来确定两个有限表示的操作数是否属于同一Wilf类。尽管如此，我们仍表明，如果一个操作员具有有限的Groebner基础，那么该操作员的单项式基础将形成无歧义的上下文无关语言。此外，我们讨论了定义语言的确定性语法。通过从Chomsky-Schutzenberger枚举定理定义的方程的代数系统中对变量进行算法消除，可以获得操作数的生成级数。然后，我们关注具有单个关系的二进制操作数的情况。该方法基于Rowland在二叉树中避免模式的结果。我们改进和完善Rowland的计算，并凭经验证实他的猜想。在这里，我们既使用代数消除方法，又使用从代数方程组直接计算形式幂级数。最后，我们讨论了Wilf类与用于算子计算的Quillen同源性的算法的联系。