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Necklaces over a group with identity product

Darij Grinberg, Peter Mao

TL;DR

This paper studies two necklace-counting problems over a finite group $\mathcal{G}$: identity-product necklaces, defined as $C_n$-orbits of $n$-tuples with product $1$, and homogeneous necklaces, as orbits under the commuting action of $C_n\times\mathcal{G}$; it proves a bijection between these two counts and derives a divisor-sum formula $|\mathcal{G}_1^n/C_n|=\frac{1}{n}\sum_{d|n} \phi\left(\frac{n}{d}\right) [\mathcal{G}\div \frac{n}{d}]\,|\mathcal{G}|^{d-1}$, with an abelian specialization. It generalizes to $K$-product necklaces and links the identity-product case to counting irreducible polynomials over finite fields via trace and normal-basis bijections, employing the orbit-counting lemma and Möbius inversion. The work unifies several counting notions (aperiodic, homogeneous, and $K$-product variants) and connects combinatorial structures to algebraic objects in finite-field theory, with applications to sequences in the OEIS. Overall, the results provide explicit divisor-sum formulas, bijections, and generalizations that illuminate the interplay between group actions and polynomial enumeration 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 $0$.

Necklaces over a group with identity product

TL;DR

This paper studies two necklace-counting problems over a finite group : identity-product necklaces, defined as -orbits of -tuples with product , and homogeneous necklaces, as orbits under the commuting action of ; it proves a bijection between these two counts and derives a divisor-sum formula , with an abelian specialization. It generalizes to -product necklaces and links the identity-product case to counting irreducible polynomials over finite fields via trace and normal-basis bijections, employing the orbit-counting lemma and Möbius inversion. The work unifies several counting notions (aperiodic, homogeneous, and -product variants) and connects combinatorial structures to algebraic objects in finite-field theory, with applications to sequences in the OEIS. Overall, the results provide explicit divisor-sum formulas, bijections, and generalizations that illuminate the interplay between group actions and polynomial enumeration 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 and a positive integer . In the first variant, we count the ``identity-product -necklaces'' -- that is, the orbits of -tuples that satisfy under cyclic rotation. In the second, we count the orbits of all -tuples under cyclic rotation and left multiplication (i.e., the operation of on given by ). We prove bijectively that both answers are the same, and express them as a sum over divisors of . Consequently, we generalize the first problem to -necklaces whose product of entries lies in a given subset of (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 .
Paper Structure (15 sections, 25 theorems, 126 equations, 3 tables)

This paper contains 15 sections, 25 theorems, 126 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Main theorem I: Identity-product $n$-necklaces
  4. Lemmas
  5. Proofs
  6. Main theorem II: Aperiodic necklaces
  7. Main theorem III: Homogeneous necklaces
  8. $K$-product necklaces
  9. Irreducible polynomials
  10. Reminders on finite fields and their Galois groups
  11. Normal bases and the bijection $\Phi$
  12. Zero-sum necklaces and zero-trace conjugacy classes
  13. The viewpoint of irreducible polynomials
  14. Sequences
  15. Further directions

Key Result

Theorem 3.1

Let $n$ be a positive integer. The number $\left| \mathcal{G}_1^n / C_n \right|$ of identity-product $n$-necklaces is Here, as usual, the summation sign means a sum over all positive divisors $d$ of $n$.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Definition 4.4
  • ...and 50 more