Related papers: Counting friezes in type D_n

We study non-zero integral friezes for Dynkin types $A_n$, $B_n$, $C_n$, $D_n$ and $G_2$. These differ from standard Coxeter-Conway (positive) friezes by allowing any non-zero integer to appear. In each case we show that there are either…

We give a short and elementary proof that every Dynkin diagram admits finitely many (positive integral) friezes. This was originally proven by Gunawan-Muller using the geometry of cluster algebras. The proof here provides an explicit…

In this article, we study infinite friezes arising from cluster categories of affine type $D$ and determine the growth coefficients for these friezes. We prove that for each affine type $D$, the friezes given by the tubes all have the same…

In this note, we show that the sequence of maximum values in frieze patterns of type $A_n$ is the sequence of Fibonacci numbers, and that of frieze patterns of type $C_n$ is the sequence of odd Fibonacci numbers.

We classify 2-periodic mesh friezes of finite type $A$, $D$ or $E$ with positive real entries. There are families with 0,1, or 2 parameters, depending on type.

The main goal of this paper is to prove several new results about frieze patterns and their equivalents, the quiddity (or $\eta$-)sequences and to obtain a formula giving the number of non-similar frieze patterns of given finite width.

In this article, we establish a link between the values of a frieze of type D and some values of a particular frieze of type A. This link allows us to compute, independently of each other, all the cluster variables in the cluster algebra…

In this article we study mutation of friezes of type $D$. We provide a combinatorial formula for the entries in a frieze after mutation. The two main ingredients in the proof include a certain transformation of a type $D$ frieze into a sub…

In this article we count tame $ SL_3 $- and $ SL_4 $-frieze patterns with width $ w $ over a finite field $ K $, as well as some tame $ SL_k $-frieze patterns for higher $ k $. Let $ n = w + k + 1 $. We consider the sets $ C_k(n) $ of…

We count numbers of tame frieze patterns with entries in a finite commutative local ring. For the ring $\mathbb{Z}/p^r\mathbb{Z}$, $p$ a prime and $r\in\mathbb{N}$ we obtain closed formulae for all heights. These may be interpreted as…

We construct frieze patterns of type D_N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram…

We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a…

Friezes patterns are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their…

Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as…

We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we…

We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\sum_{d|n} \phi(n/d)\binom{2d}{d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the…

We consider the variant of Coxeter-Conway frieze patterns called 2-frieze. We prove that there exist infinitely many closed integral 2-friezes (i.e. containing only positive integers) provided the width of the array is bigger than 4. We…

In this article, we construct SL$_k$-friezes using Pl\"ucker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Pl\"ucker embedding. When this cluster…

We determine the number of positive integral points on $n$-dimensional affine varieties associated to arbitrary $n \times n$ generalized Cartan matrices. An application to the theory of cluster algebras and combinatorics is the resolution…

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type An. Such diamonds constitute a tool to build integral frieze patterns.