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.
Highlights
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 use Lindstrom-GesselViennot methodology to find the number of tilings of a quartered lozenge hexagon
A lattice divides the plane into fundamental regions
A tile is the union of any two fundamental regions sharing an edge
Summary
A lattice divides the plane into fundamental regions. A (lattice) region is a finite connected union of fundamental regions of that lattice. Aztec diamond of order n by removing the squares running along the northwestern and northeastern sides, denoted by T An. Label the squares on the southwestern and southeastern sides of ADn and T An by. We consider the trimmed Aztec rectangle region T Rm,n obtained from ARm,n by removing squares running along its northwestern and northeastern sides (see Figure 1.3(b)). Similar to quartered Aztec diamonds, we consider the region obtained from ARm,n by removing even squares on the southwestern side, and removing arbitrarily n. The numbers of tilings of quartered Aztec rectangles are given by simple product formulas involving two functions E(...) and O(...) defined in (1.1) and (1.2). We remove all squares along the southeastern side of the Aztec rectangle, and remove the bottommost square the resulting region.
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