I'm currently reading Nieper-Wißkirchen's Chern numbers and Rozansky-Witten invariants of compact Hyper-Kähler manifolds. He introduces the space of graph homology $\mathcal B$ as the free $K[\circ]$-module generated by Jacobi diagrams (unitrivalent graphs) modulo the two relations AS (anti-symmetry) and IHX. There is a cup-product $\cup: \mathcal B \otimes \mathcal B \to \mathcal B$ given by disjoint union of graphs.
Nieper-Wißkirchen also defines a second product $\times: \mathcal B\otimes \mathcal B\to \mathcal B$, given by the following procedure: Given a graph $\gamma$ with $n$ legs, define $s(\gamma)$ by choosing a labeling of the legs which gives $\hat \gamma \in \mathcal B_{\{1,\dotsc,n\}}$, and set $$s(\gamma) = \frac 1 {n!} \sum_{\sigma \in S_n} \sigma_* \hat \gamma.$$ Here $S_n$ is the symmetric group and $\sigma_* \hat \gamma$ is obtained by relabeling $\hat \gamma$. Clearly $s(\gamma)$ does not depend on the choice of $\hat \gamma$. Given two graphs $\gamma, \gamma'$ with $n$ and $n'$ legs, the product $\gamma \times \gamma'$ is defined as the image of the element $$s(\gamma) \cup s(\gamma') \in \mathcal B_{\{1,\dotsc, n\} \sqcup \{1, \dotsc, n'\}}$$ under the map $\mathcal B_{\{1,\dotsc, n\} \sqcup \{1, \dotsc, n'\}} \to \mathcal B$ which forgets the labeling.
I was wondering how this new product $\times$ differs from $\cup$? I think that $$s(\gamma) \cup s(\gamma') = \frac 1 {n!(n')!} \sum_{\sigma \in S_n} \sum_{\tau \in S_{n'}} \sigma_* \hat \gamma \cup \tau_* \hat \gamma'$$ contains $n! (n')!$ many summands, each of which is basically $\gamma \cup \gamma'$ with some labeling. So if one forgets the labeling the sum collapses and just gives $\gamma \times \gamma' = \gamma \cup \gamma'$?
This seems to be a misunderstanding, but I can't see my mistake.
(I tagged this question "knot-theory" because I realized that trivalent graphs, the space of graph homology and similar products appear in the context of knots, though I don't know anything about this).
