Related papers: Combinatorial connections in snake graphs: Tilings…
Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we continue our…
We study cluster algebra of affine type $A_1^{(1)}$ by using two methods including counting the numbers of perfect matchings on snake graphs and compatible pairs on maximal Dyck paths. We find that the sum of coefficients of the terms in…
Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we identify…
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
This paper is a slightly extended version of the talk I gave at the Open Problems in Algebraic Combinatorics conference at the University of Minnesota in May 2022. We introduce two strict order relations on lattice paths and formulate…
The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyn's and Temperley--Fisher's ideas about spectral properties of weighted adjacency matrices of planar…
Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake…
For cluster algebras of surface type, Musiker, Schiffler and Williams gave a formula for cluster variables in terms of perfect matchings of snake graphs. Building on this, we provide a simple determinantal formula for cluster variables via…
This paper establishes a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion w can be associated to a piecewise-linear poset P and a snake…
We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that…
The perfect matching complex of a simple graph $G$ is a simplicial complex having facets (maximal faces) as the perfect matchings of $G$. This article discusses the perfect matching complex of polygonal line tilings and the $\left(2 \times…
We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by…
It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…
A pure snake configuration is a bijection $\sigma:\mathbb{Z}^2 \to \mathbb{Z}^2$ containing no two-cycles and such that for each $x \in \mathbb{Z}^2$ we have $\sigma(x) \in \{ x , x+ \mathbf{e}^1, x+\mathbf{e}^2 , x- \mathbf{e}^2 \}.$ The…
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To…
In preference modelling, it is essential to determine the number of questions and their arrangements to ask from the decision maker. We focus on incomplete pairwise comparison matrices, and provide the optimal filling in patterns, which…
In this paper we present a new version of the second author's factorization theorem for perfect matchings of symmetric graphs. We then use our result to solve four open problems of Propp on the enumeration of trimer tilings on the hexagonal…
We continue our study of orderings on Bratteli diagrams started in previous work, joint with Jan Kwiatkowski, where Bratteli diagrams of finite rank were considered. We extend the notions of languages, permutations (called correspondences…
We give a combinatorial intepretation of cluster variables of a specific cluster algebra under a mutation sequence of period 6, in terms of perfect matchings of subgraphs of the brane tiling dual to the quiver associated with the cluster…
Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer…