A bijection between necklaces and multisets with divisible subset sum

TL;DR

This paper establishes a bijection between necklaces with limited colors and multisets with divisible subset sums, revealing a deep combinatorial connection and solving a problem posed by Richard Stanley.

Contribution

It introduces a new bijection between necklaces and multisets with divisible subset sums, especially when parameters are coprime or prime powers, and addresses a problem by Stanley.

Findings

Cardinality equivalence for coprime q and n

Explicit bijection for prime power q

Resolution of Stanley's bijective problem for q=2

Abstract

Consider these two distinct combinatorial objects: (1) the necklaces of length $n$ with at most $q$ colors, and (2) the multisets of integers modulo $n$ with subset sum divisible by $n$ and with the multiplicity of each element being strictly less than $q$. We show that these two objects have the same cardinality when $q$ and $n$ are mutually coprime. Additionally, when $q$ is a prime power, we construct a bijection between these two objects by viewing necklaces as cyclic polynomials over the finite field of size $q$. Specializing to $q=2$ answers a bijective problem posed by Richard Stanley.