Suppose I have a symmetric operad $\mathcal{P}$ defined over $\text{Vect}_{\mathbb{K}}$ with generators and relations in degrees at most $l$. Now suppose I already know $\mathcal{P}(k)$ as an $S_k$-representation for $0 \leq k \leq l$.
How do I determine $\mathcal{P}(l+1)$ from $\mathcal{P}(0), \mathcal{P}(1), \ldots, \mathcal{P}(l)$ ? More generally, how do I determine $\mathcal{P}(k)$ for some large $k$?
My current technique is to find vector space generators for each grade of the operadic ideal of relations by building trees in which some relation appears. Then I use linear algebra and Schur-Weyl duality to determine each grade of the ideal as a representation. This technique is reliable for small cases, but the number of trees grows large and this generating set is usually quite redundant.
I am hoping (perhaps naively) for some combinatorial rule in terms of Young diagrams which can answer the question explicitly.
Edit: Vladimir Dotsenko suggests that I limit the scope, so here's a (more reasonable) question which is also interesting:
Assume further that $\mathcal{P}$ is generated by a single associative binary operation. Now can we determine $\mathcal{P}(k)$ from an initial segment using the combinatorics of Young diagrams?
It seems like the Littlewood-Richardson rule might be of some help here. There also seems to be some plethysm going on. Perhaps these tools are sufficient?
