## Abstract

Given an operad P with a finite Gröbner 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.

Original language | English |
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Pages (from-to) | 377-429 |

Number of pages | 53 |

Journal | Journal of Algebra |

Volume | 426 |

DOIs | |

State | Published - 5 Mar 2015 |

Externally published | Yes |

### Bibliographical note

Funding Information:The first author's research is partially supported by NSh-1500.2014.2, by RFBR grants 13-02-00478 , 13-01-12401 , 15-01-09242 , by “The National Research University–Higher School of Economics” Academic Fund Program in 2013–2014 , research grant 14-01-0124 , by Dynasty Foundation and Simons-IUM fellowship . The second author's research was supported by “The National Research University Higher School of Economics Academic Fund Program” in 2013–2014 , research grant 12-01-0134 , and the RFBR project 14-01-00416 .

Publisher Copyright:

© 2014 Elsevier Inc..

## Keywords

- Algebraic differential equation
- Generating series
- Gröbner basis for operads
- Operad
- Variety of algebras

## ASJC Scopus subject areas

- Algebra and Number Theory