Necklaces over a group with identity product
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912786112577536 |
|---|---|
| author | Grinberg, Darij Mao, Peter |
| author_facet | Grinberg, Darij Mao, Peter |
| contents | 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 $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08937 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Necklaces over a group with identity product Grinberg, Darij Mao, Peter Combinatorics Group Theory 05E18, 05A15, 11T06 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 $0$. |
| title | Necklaces over a group with identity product |
| topic | Combinatorics Group Theory 05E18, 05A15, 11T06 |
| url | https://arxiv.org/abs/2405.08937 |