Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1802.03507 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.