Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4655350 | Journal of Combinatorial Theory, Series A | 2014 | 13 Pages |
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−1d−1, and the elementary divisors only take values d−1d−1 and 1. We also show that in the generalized frieze patterns obtained in our setting every adjacent 2×22×2-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d=3d=3 gives back the Conway–Coxeter condition on frieze patterns.