Domino Shuffling on Novak Half-Hexagons and Aztec Half-Diamonds

Alternating-Sign Matrices and Domino Tilings (Part I)

We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2()/ domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and ...

A Note on Domino Shuffling

We present a variation of James Propp’s generalized domino shuffling, which provides an efficient way to obtain perfect matchings of weighted Aztec diamonds. Our modification is specially tailored to deal with cases when some of the weights are zero. This allows us to tile efficiently a large class of planar graphs, by embedding them in a large enough Aztec diamond. We also give a sufficient co...

Computing a pyramid partition generating function with dimer shuffling

Abstract. We verify a recent conjecture of Kenyon/Szendrői by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson–Thomas theory of a non-commutative resolution of the conifold singularity {x1x2 −x3x4 = 0} ⊂ C. The proof does not require algebraic geo...

Enumeration of Tilings of Diamonds and Hexagons with Defects

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to open problems 1, 2, and 10 in James Propp’s list of problems on enumeration of matchings [22].

On Domino Tilings of Rectangles Aztec Diamonds in the Rough