In this article, Temperley’s bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spann...

We prove log-concavity of exit probabilities lattice random walks in certain planar regions.

We investigate properties which hold for both the lattice of a binary relation and for its ’mirror lattice’, which is the lattice of the complement relation. We first prove that the relations whose lattice is dismantlable correspond to the class of chordal bipartite graphs; we provide algorithmic tools to find a doubly irreducible element in such a lattice. We go on to show that a lattice is di...

The notion of CD-independence is introduced as follows. A subset X of a lattice L with 0 is called CD-independent if for any x, y ∈ X , either x ≤ y or y ≤ x or x ∧ y = 0. In other words, if any two elements of X are either Comparable or Disjoint. Maximal CD-independent subsets are called CD-bases. The main result says that any two CD-bases of a finite distributive lattice L have the same numbe...