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ENUMERATING NECKLACES WITH TRANSITIONS

Part of: Graph theory Combinatorics

Published online by Cambridge University Press:  11 May 2021

FRANCESCO BIANCONI Open the ORCID record for FRANCESCO BIANCONI [Opens in a new window]  and
EMANUELE BRUGNOLI Open the ORCID record for EMANUELE BRUGNOLI [Opens in a new window]
FRANCESCO BIANCONI*
Affiliation:
Department of Engineering, Università degli Studi di Perugia, Via Goffredo Duranti 93, 06125 Perugia, Italy
EMANUELE BRUGNOLI
Affiliation:
Communications Regulatory Authority (AGCOM), Centro Direzionale Isola B5, Naples, Italy e-mail: brugnoliema@gmail.com
*
e-mail: bianco@ieee.org
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Abstract

Necklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

Keywords

colouringcompositionnecklacetransition

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 1 - 11
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Partially supported by the Department of Engineering, Università degli Studi di Perugia, Italy, within the project Artificial Intelligence for Earth Observation (Fundamental Research Grant Scheme 2020).

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