On generating series of finitely presented operads

Details On generating series of finitely presented operads On generating series of finitely presented operads

2012-02-23

arxiv.org/abs/1202.5170v1

Mathematics

Quantum Algebra

preliminary draft, comments are welcome

Scientific paper

Given an operad P with a finite Gr\"obner 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 for P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions 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.

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