Enumeration of Tilings of Quartered Aztec Rectangles

Abstract

We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindström-Gessel-Viennot methodology to find the number of tilings of a quartered lozenge hexagon.

Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary

“…This section is devoted to the proof of Lemma 1.4, based on the previous result of the author in [17] about a family of regions called quartered hexagons as follows.…”

“…, a k ) in [17]. However, the work in [17] does not cover the last equality (3.4) (in particular, the region QH 2k−1,n (a 1 , a 2 , . .…”

“…In his generalization of W. Jockusch and J. Propp's result on quartered Aztec diamonds [16], the author [17] investigated a family of regions, called quartered hexagons, that is obtained from a quarter of a symmetry hexagon by removing several unit triangles from the base (see the regions in Figure 3.1). We also notice that the tiling enumeration of the quartered hexagons in [17] is equivalent to the lattice-path enumeration of the so-called stars by C. Krattenthaler, A. J. Guttmann, and X. G. Viennot [13]. This result implies the tiling enumeration of the following halved hexagons with triangles removed from the base.…”

“…In addition to the above 'natural' reduction, the extreme case of x = 0 is also interesting, since it gives a reduction to a product of the tiling numbers of two quartered hexagons in [17] (see Figure Figure 4.1), i.e., for odd y (1.13) M(R 0,y,z (a 1 , a 2 , . .…”

A new proof for the number of lozenge tilings of quartered hexagons

“…Theorem 1.1 (Proctor [6]). Assume that a, b, c are non-negative integer so that b ≥ c. Let P a,b,c be the region obtained from the hexagon H a,b,c by removing the "maximal staircase" from its east corner (see Figure 1.1 for P 3,6,4 ). Then…”