Abstract
Much work has been done to count the number of domino tilings for Aztec diamonds and augmented Aztec diamonds. Augmented Aztec rectangles and their chains are generalizations of these shapes. In this paper, we use Delannoy paths to count the number of domino tilings for these rectangles and their chains.
Highlights
The Aztec diamond of order n is the union of unit squares with integral corners (x, y) satisfying |x| + |y| n + 1 in the Cartesian coordinate system of R2
A domino tiling of a region is a set of non-overlapping dominoes covering the region
The enumeration of domino tilings for the Aztec diamond is a problem of rich content in both enumerative combinatorics and statistical mechanics
Summary
The Aztec diamond of order n is the union of unit squares with integral corners (x, y) satisfying |x| + |y| n + 1 in the Cartesian coordinate system of R2. The enumeration of domino tilings for the Aztec diamond is a problem of rich content in both enumerative combinatorics and statistical mechanics. The number of domino tilings for the augmented Aztec diamond of order n is shown by Sachs and Zernitz [14] to be n k=0 n k.
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