A Bijection between Necklaces and Restricted Multisets

TL;DR

This paper proves a conjecture by Swee Hong Chan, establishing a bijection between necklaces with limited colors and certain periodic functions with divisible weighted sums, revealing a deep combinatorial connection.

Contribution

It provides a rigorous proof of Chan's conjecture, linking necklaces and restricted multisets through a novel bijection.

Findings

Established a bijection between necklaces and periodic functions

Connected combinatorial objects with divisibility conditions

Confirmed conjecture for coprime integers n and q

Abstract

We present a proof of Swee Hong Chan's conjecture establishing a bijection between the set of necklaces of length $n$ with at most $q$ colors, and the set of periodic functions $f: \mathbb{Z}_{n}\to {0, 1, ..., q-1}$ whose weighted sum is divisible by $n$, where $q$ and $n$ are coprime positive integers.