---
tag: BJH5SAJ8Y9
kind: concept
name: Necklace polynomial
aliases: [moreau-necklace-counting-function, moreaus-necklace-counting-function, necklace-counting-function]
sources:
  - wikipedia: en:430792@r1363307816
    locator: Necklace polynomial
    note: adopted
    mention: wikipedia://en/430792
    sections: [Summary]
same_as:
  wikidata: Q6985582
  wikipedia: Necklace polynomial
inventory: [8NPABJA0BJ]
---

## Summary

In [[combinatorial|0JKW7W7XFR]] mathematics, the **necklace polynomial**, or **Moreau's necklace-counting function,** introduced by (txt), counts the number of distinct necklaces of $n$ colored beads chosen out of $k$ available colors, arranged in a cycle. Unlike the usual problem of [[graph coloring|038YGY0HSC]], the necklaces are assumed to be aperiodic (not composed from a repeated subsequence), and counted up to rotation (rotating the beads around the necklace counts as the same necklace), but without flipping over (reversing the order of the beads counts as a different necklace). This counting function also describes the dimensions in a free Lie algebra and the number of irreducible polynomials over a finite field.