The multisubset sum problem for finite abelian groups

TL;DR

This paper derives explicit formulas for counting multisubsets and partitions of elements in finite abelian groups with specified sizes and sums, providing tools for combinatorial enumeration in algebraic structures.

Contribution

It introduces explicit formulas and inclusion-exclusion methods for counting multisubsets and partitions in finite abelian groups, advancing combinatorial enumeration techniques.

Findings

Explicit formula for multisubsets with a given sum

Counting partitions of an element into parts over a finite abelian group

Inclusion-exclusion formula for multisubsets with constraints

Abstract

In this note, we give the explicit formula for the number of multisubsets of a finite abelian group $G$ with any given size such that the sum is equal to a given element $g\in G$. This also gives the number of partitions of $g$ into a given number of parts over a finite abelian group. An inclusion-exclusion formula for the number of multisubsets of a subset of $G$ with a given size and a given sum is also obtained.