A new sieve for restricted multiset counting

TL;DR

This paper extends the Li--Wan sieve to multisets with symmetric underlying sets, using Mobius inversion on partition posets, and applies it to problems in finite fields and zero-sum multisets over cyclic groups.

Contribution

It introduces a new sieve method for restricted multiset counting, expanding the applicability of the Li--Wan sieve to symmetric multisets.

Findings

Extended sieve applicable to symmetric multisets

Derived formulas for multiset partition problems over finite fields

Analyzed zero-sum multiset problems in cyclic groups

Abstract

The Li--Wan sieve is extended to multisets when the underlying set is symmetric. The main ingredient of the proof is the Mobius inversion formula on the poset of partitions of $\{1,2,\dots,k\}$ ordered by refinement. As illustrative applications, we investigate the problems of partitions over finite fields and zero-sum multisets over the additive group $\mathbb{Z}/n\mathbb{Z}$. .