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Let \begin{equation} G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q) \end{equation} be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net description of free PROs).

Let us denote by $H_G(x, y, z)$ the generating series of $\mathcal{F}(G)$, defined by \begin{equation} H_G(x, y, z) := \sum_{p \geq 0} \: \sum_{q \geq 0} \: \sum_{e \in \mathcal{F}(G)(p, q)} x^p y^q z^{\deg e}, \end{equation} where $\deg e$ is the degree of $e \in \mathcal{F}(G)$, that is the number of generators of $G$ needed to build $e$. In other words, $H_G(x, y, z)$ is the generating series that counts the elements of $\mathcal{F}(G)$ by the input arity (parameter $x$), by the output arity (parameter $y$), and by the degree (parameter $z$).

My question is the following:

Given a set $G$ of generators, do we know an expression for the generating series $H_G$ for $\mathcal{F}(G)$?

I think, since this question is very basic and natural, that it would appear in some papers dealing with PROs. References are thus welcome.

[1] Y. Lafont, Diagram rewriting and operads, Operads 2009, ed. J.-L. Loday & B. Vallette, Séminaires et Congrès 26, p. 163-179, SMF, 2011.

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