Counting packings of generic subsets in finite groups

TL;DR

This paper derives a formula for counting packings of generic subsets in finite groups, generalizing falling factorials to account for larger subsets and their disjoint arrangements.

Contribution

It introduces a formula for the number of packings of generic subsets in finite groups, extending classical combinatorial counts to more complex set configurations.

Findings

Provides a general counting formula for packings in finite groups

Extends falling factorials to subsets of arbitrary size

Enables enumeration of disjoint subset arrangements in algebraic structures

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.