Counting Packings of Generic Subsets in Finite Groups

A packing of subsets $\mathcal S_1,\dots,\mathcal S_n$ in a group $G$ is an element $(g_1,\dots,g_n)$ of $G^n$ such that $g_1\mathcal S_1,\dots,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is finite and if the subsets $\mathcal S_1,\dots,\mathcal S_n$ satisfy a genericity condition. This formula can be seen as a generalization of the falling factorials which encode the number of packings in the case where all the sets $\mathcal S_i$ are singletons.