On generating series of finitely presented operads

TL;DR

This paper investigates the generating functions of finitely presented operads with finite Groebner bases, proving their algebraic or rational nature under certain growth conditions, and providing examples and conjectures.

Contribution

It establishes conditions under which the generating functions of operads are algebraic or rational, extending understanding of operad growth and structure.

Findings

Exponential generating functions are differential algebraic, algebraic if operad is symmetrized.

Ordinary generating functions are rational under polynomial or exponential growth bounds.

Provides examples and discusses conjectures for broader classes of operads.

Abstract

Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.