Counting packings of generic subsets in finite groups

Abstract

A packing of subsets ${\mathcal{S}}_1,...,{\mathcal{S}}_n$ in a group $G$ is a sequence $(g_1,...,g_n)$ such that $g_1{\mathcal{S}}_1,...,g_n{\mathcal{S}}_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is finite and if the subsets ${\mathcal{S}}_1,...,{\mathcal{S}}_n$ 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.