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Preprint
Necklaces over a group with identity product
arXiv.org
14 May 2024
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 $n$.
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Details
- Title
- Necklaces over a group with identity product
- Creators
- Darij GrinbergPeter Mao
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021878815604721