Let $B(n)$ be a set of leaf-labelled inequivalent binary rooted trees. $S_n$ be a symetric group. We say that $\gamma\in B(n)$ fixed by some $\sigma\in S_n$, if $\sigma(\gamma)=\gamma$ . Here is an example in the picture enter image description here.
The number of leaf-labelled rooted binary trees fixed by $\sigma$ is calculated by the given formula: enter image description here, where $\lambda$ is a partition type of $\sigma$. For instance type of $(12)(3)(4)(5)\in S_5$ is $(2,1,1)$, $(1234)(56)(7)\in S_7$ is $(4,2,1).$
In the anti-commutative case I do not have an idea how to find the number of $\gamma\in B(n)$ fixed by some $\sigma\in S_n$. Can someone give an idea where can I start. By the context "anti-commutative case" I mean, for some $\gamma\in B(n)$, we can have $\sigma(\gamma)=-\gamma$.