Enumeration of Tilings of Quartered Aztec Rectangles

Alternating sign matrices and tilings of Aztec rectangles

The problem of counting numbers of tilings of certain regions has long interested researchers in a variety of disciplines. In recent years, many beautiful results have been obtained related to the enumeration of tilings of particular regions called Aztec diamonds. Problems currently under investigation include counting the tilings of related regions with holes and describing the behavior of ran...

A Simple Proof for the Number of Tilings of Quartered Aztec Diamonds

In this paper a (lattice) region is a connected union of unit squares in the square lattice. A domino is the union of two unit squares that share an edge. A (domino) tiling of a region R is a covering of R by dominos such that there are no gaps or overlaps. Denote by T(R) the number of tilings of the region R. The Aztec diamond of order n is defined to be the union of all the unit squares with ...

Applications of graphical condensation for enumerating matchings and tilings

A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in o...

On Domino Tilings of Rectangles Aztec Diamonds in the Rough

A Gessel-Viennot-Type Method for Cycle Systems in a Directed Graph

We introduce a new determinantal method to count cycle systems in a directed graph that generalizes Gessel and Viennot’s determinantal method on path systems. The method gives new insight into the enumeration of domino tilings of Aztec diamonds, Aztec pillows, and related regions.