On the number of antipodal bicolored necklaces
type: Item
instance of: scholarly article
title: On the number of antipodal bicolored necklaces
zbMATH Open document ID: 1295.11015
author: Florian Luca
author: Ricardo Strausz
published in: Aequationes Mathematicae
publication date: 2012-02-22T00:00:00Z
review text: An \textit{antipodal bicolored necklace} is a congruence class modulo the dihedral group of the set of colorings of the vertices of a regular \((2n)\)-gon with two colors, such that antipodal vertices receive different colors. In [\textit{J. J. Montellano-Ballesteros} and \textit{R. Strausz}, J. Comb. Theory, Ser. A 106, No. 1, 109--121 (2004; Zbl 1042.05024)], it was proved that the number \(\nu(2n)\) of antipodal bicolored necklaces is related to the number of combinatorial types of \((n-3)\)-dimensional convex polytopes with \(n\) vertices in general position, of which there are \(\nu(2n) - \lceil {n \over 2} \rceil\). As a consequence, \(\nu(2n)\) can be explicitly computed when \(n\) is a power of 2 or when \(n\) is an odd prime. The main result of the paper under review states that the sequence \((\nu(2n))_{ n \geq 1 }\) is non-holonomic, i.e., it does not satisfy a linear recurrence of a fixed finite order with polynomial coefficients. The authors derive this result by finding the first two terms of the asymptotic expansion of~\(\nu(2n)\).
reviewed by: Matthias Beck
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 6008792
zbMATH Keywords: antipodal bicolored necklaces
zbMATH Keywords: non-holonomic sequences
MaRDI profile type: Publication
full work available at URL: s00010-010-0067-5
OpenAlex ID: W2027401481
cites work: Non-holonomicity of sequences defined via elementary functions
cites work: Lindelöf representations and (non-)holonomic sequences
cites work: Catalan and Apéry numbers in residue classes
cites work: Q3530209
cites work: Witt vectors and the algebra of necklaces
cites work: Counting polytopes via the Radon complex
DOI: 10.1007/S00010-010-0067-5
description: scientific article; zbMATH DE number 6008792