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Generalized frieze pattern determinants and higher angulations of polygons

Lookup NU author(s): Professor Peter Jorgensen

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Abstract

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d-angulation, the determinant is a power of d - 1, and the elementary divisors only take values d - 1 and 1. We also show that in the generalized frieze patterns obtained in our setting every ajacent 2 x 2-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d = 3 gives back the Conway-Coxeter condition on frieze patterns. (C) 2013 Elsevier Inc. All rights reserved.

Publication metadata

Author(s): Bessenrodt C, Holm T, Jorgensen P

Publication type: Article

Publication status: Published

Journal: Journal of Combinatorial Theory, Series A

Year: 2014

Volume: 123

Issue: 1

Pages: 30-42

Print publication date: 25/11/2013

ISSN (print): 0097-3165

ISSN (electronic): 1096-0899

Publisher: Academic Press

URL: http://dx.doi.org/10.1016/j.jcta.2013.11.003

DOI: 10.1016/j.jcta.2013.11.003

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