Necklaces over a group with identity product

TL;DR

This paper studies two variants of necklace counting problems over finite groups, establishing their equivalence, deriving formulas, and extending to subsets of the group, with applications to polynomial enumeration over finite fields.

Contribution

It provides a bijective proof linking two necklace counting variants, generalizes to subsets of groups, and connects to finite field polynomial enumeration.

Findings

Both necklace counting variants are proven to be the same.

Formulas for counting necklaces are expressed as sums over divisors of n.

Connections are made to counting irreducible polynomials over finite fields.

Abstract

We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product $n$-necklaces'' -- that is, the orbits of $n$-tuples $\left(a_1, a_2, \ldots, a_n\right) \in \mathcal{G}^n$ that satisfy $a_1 a_2 \cdots a_n = 1$ under cyclic rotation. In the second, we count the orbits of all $n$-tuples $\left(a_1, a_2, \ldots, a_n\right) \in \mathcal{G}^n$ under cyclic rotation and left multiplication (i.e., the operation of $\mathcal{G}$ on $\mathcal{G}^n$ given by $h \cdot \left(a_1, a_2, \ldots, a_n\right) = \left(ha_1, ha_2, \ldots, ha_n\right)$). We prove bijectively that both answers are the same, and express them as a sum over divisors of $n$. Consequently, we generalize the first problem to $n$-necklaces whose product of entries lies in a given subset of $\mathcal{G}$ (closed under conjugation), and we connect a particular case to the enumeration of irreducible polynomials over a finite field with given degree and second-highest coefficient $0$.