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arXiv · 1305.1098

Generalized frieze pattern determinants and higher angulations of polygons

Christine Bessenrodt·Thorsten Holm·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 adjacent 2x2-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.

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Christine Bessenrodt, Thorsten Holm, Peter Jorgensen. 2013-05-06. Generalized frieze pattern determinants and higher angulations of polygons. https://doi.org/10.1016/j.jcta.2013.11.003

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