Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 30-42 |
| Seitenumfang | 13 |
| Fachzeitschrift | Journal of Combinatorial Theory. Series A |
| Jahrgang | 123 |
| Ausgabenummer | 1 |
| Frühes Online-Datum | 25 Nov. 2013 |
| Publikationsstatus | Veröffentlicht - 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.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Theoretische Informatik
- Mathematik (insg.)
- Diskrete Mathematik und Kombinatorik
- Informatik (insg.)
- Theoretische Informatik und Mathematik
Zitieren
- Standard
- Harvard
- Apa
- Vancouver
- BibTex
- RIS
in: Journal of Combinatorial Theory. Series A, Jahrgang 123, Nr. 1, 04.2014, S. 30-42.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › 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 -