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 …

We study cluster categories arising from marked surfaces (with punctures and non-empty
boundaries). By constructing skewed-gentle algebras, we show that there is a bijection …

We prove a conjecture of Morier-Genoud and Ovsienko that says that rank polynomials of
the distributive lattices of lower ideals of fence posets are unimodal. We do this by proving a …

We present a new and very concrete connection between cluster algebras and knot theory.
This connection is being made via continued fractions and snake graphs. It is known that the …

In this article, we realize skew-gentle algebras as skew-tiling algebras associated to
admissible partial triangulations of punctured marked surfaces. Based on this, we establish …

This paper is a sequel to our previous work in which we found a combinatorial realization of
continued fractions as quotients of the number of perfect matchings of snake graphs. We …

We express cluster variables of type B n and C n in terms of cluster variables of type A n.
Then we associate a cluster tilted bound symmetric quiver Q of type A 2 n− 1 to any seed of …

In this paper we introduce abstract string modules and give an explicit bijection between the
submodule lattice of an abstract string module and the perfect matching lattice of the …

Markov numbers are integers that appear in the solution triples of the Diophantine equation,
x 2+ y 2+ z 2= 3 xyz, called the Markov equation. A classical topic in number theory, these …

The Markov numbers are the positive integers that appear in the solutions of the equation x
2+ y 2+ z 2= 3 xy z. These numbers are a classical subject in number theory and have …