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On generating series of finitely presented operads
Cornell University
,
2012.
No. 1202.5170.
With any given operad $\mathcal{P}=\cup_{i=1}^{\infty}\mathcal{P}(n)$ we can associate a generating series of dimensions of the space of operations with the same arity. This article is an attempt to find a reasonable bounds and recursive relations for these generating series. Of course, for arbitrary operad the corresponding series may be transcendental, therefore we restrict our self to the case of operads that admits a finite Grobner basis. Recall, that there exists the theory of monomials and Grobner bases for nonsymmetric operads and there is no corresponding theory for symmetric operads. In order to avoid this problem one has to forget part of the action of symmetric group. The latter theory is called shuffle operads and is described in my joint <a href="Grobner">paper</a>. The generating series of an operad with a chosen finite Grobner basis and a generating series of the associated graded operad with monomial relations are the same. Therefore, the problem we consider reduces to the description of generating series of monomial operads. The main result of this note looks as follows: The ordinary generating series $\sum_{n=1}^{\infty} dim{\mathcal{P}}(n) t^n$ of a finitely presented monomial nonsymmetric operad ${\mathcal{P}}$ is an algebraic function. The exponential generating series $\sum_{n=1}^{\infty} dim{\mathcal{P}}(n) \frac{t^{n}}{n!}$ of a finitely presented monomial shuffle operad ${\mathcal{P}}$ is differential algebraic function if the set of relations is closed under shuffle-permutations. The shuffle-permutations are those permutations which permutes the labels of a planar tree-monomial, but do not change the underlying planar tree. The proofs are constructive. Namely, we present algorithms on how to find a finite system of recursive equations. The positivity of the coefficients in these systems implies some restrictions on the generating series with small growths: If (in addition to aforementioned finiteness assumptions) the growth of the dimensions $\mathcal{P}(n)$ 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 $\{ dim {\mathcal{P}}(n) \}$ is rational. All algorithms are provided by series of examples. Being inspired by hunting examples out of the PBW case we have discovered several non-quadratic examples that have their own interest.