Details
| Original language | English |
|---|---|
| Pages (from-to) | 30-42 |
| Number of pages | 13 |
| Journal | Journal of Combinatorial Theory. Series A |
| Volume | 123 |
| Issue number | 1 |
| Early online date | 25 Nov 2013 |
| Publication status | Published - Apr 2014 |
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 2 × 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.
Keywords
- Determinant, Elementary divisor, Frieze pattern, Polygon, Smith normal form, Symmetric matrix
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Computer Science(all)
- Computational Theory and Mathematics
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In: Journal of Combinatorial Theory. Series A, Vol. 123, No. 1, 04.2014, p. 30-42.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Generalized frieze pattern determinants and higher angulations of polygons
AU - Bessenrodt, Christine
AU - Holm, Thorsten
AU - Jørgensen, Peter
N1 - Funding Information: This work has been carried out in the framework of the research priority programme SPP 1388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft (DFG) . We gratefully acknowledge financial support through the grant HO 1880/5-1 .
PY - 2014/4
Y1 - 2014/4
N2 - 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 2 × 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.
AB - 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 2 × 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.
KW - Determinant
KW - Elementary divisor
KW - Frieze pattern
KW - Polygon
KW - Smith normal form
KW - Symmetric matrix
UR - http://www.scopus.com/inward/record.url?scp=84888085597&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2013.11.003
DO - 10.1016/j.jcta.2013.11.003
M3 - Article
AN - SCOPUS:84888085597
VL - 123
SP - 30
EP - 42
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
SN - 0097-3165
IS - 1
ER -