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 …

The set of perfect matchings of a connected bipartite plane graph $ G $ has the structure of a
distributive lattice, as shown by Propp, where the partial order is induced by the height of a …

q-DEFORMED RATIONALS AND q-CONTINUED FRACTIONS Page 1 Forum of Mathematics,
Sigma (2020), Vol. 8, e13, 55 pages doi:10.1017/fms.2020.9 1 q-DEFORMED RATIONALS …

Define an expansion poset to be the poset of monomials of a cluster variable attached to an
arc in a polygon, where each monomial is represented by the corresponding combinatorial …

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 …

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 …

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles
(squares) T 1, T 2,…, T n. In this case, for 1≤ j≤ n− 1, two consecutive tiles T j and T j+ 1 …

Recently, Çanakçi and Schroll proved that associated with a string module $ M (w) $ there is
an appropriated snake graph $\mathscr {G} $. They established a bijection between the …

The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows,
one source and one sink (ie, a quiver with two vertices and two arrows going in the same …