On generating series of finitely presented 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.

Minor changes; references to recent articles by Berele and by Belov, Bokut, Rowen, and Yu are added

References in corpus (4)

• Quillen homology for operads via Gröbner bases
• The Jacobian Conjecture, together with Specht and Burnside-type problems
• Pattern avoidance in labelled trees
• Algebraic Properties of Codimension Series of PI-Algebras

Cited by in corpus (14)

• Gröbner methods for representations of combinatorial categories
• The Jacobian Conjecture, together with Specht and Burnside-type problems
• Nonsymmetric operads in combinatorics
• Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras
• Tree series and pattern avoidance in syntax trees
• Truncation of Unitary Operads
• Growth of nonsymmetric operads
• Growth in varieties of multioperator algebras and Groebner bases in operads
• Associator dependent algebras and Koszul duality
• Quotients of the magmatic operad: lattice structures and convergent rewrite systems
• Non-Contiguous Pattern Avoidance in Binary Trees
• The combinator $\mathbf{M}$ and the Mockingbird lattice
• Wilf classes of non-symmetric operads
• Polynomial automorphisms, quantization and Jacobian conjecture related problems