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Full-Text Articles in Physical Sciences and Mathematics
Signed Lozenge Tilings, D. Cook Ii, Uwe Nagel
Signed Lozenge Tilings, D. Cook Ii, Uwe Nagel
Mathematics Faculty Publications
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a …
Enumeration Of Tilings Of Quartered Aztec Rectangles, Tri Lai
Enumeration Of Tilings Of Quartered Aztec Rectangles, Tri Lai
Department of Mathematics: Faculty Publications
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 Lindstr¨om-Gessel- Viennot methodology to find the number of tilings of a quartered lozenge hexagon.
A Generalization Of Aztec Diamond Theorem, Part I, Tri Lai
A Generalization Of Aztec Diamond Theorem, Part I, Tri Lai
Department of Mathematics: Faculty Publications
We consider a new family of 4-vertex regions with zigzag boundary on the square lattice with diagonals drawn in. By proving that the number of tilings of the new regions is given by a power 2, we generalize both Aztec diamond theorem and Douglas’ theorem. The proof extends an idea of Eu and Fu for Aztec diamonds, by using a bijection between domino tilings and non-intersecting Schr¨oder paths, then applying Lindstr¨om-Gessel-Viennot methodology.
A Simple Proof For The Number Of Tilings Of Quartered Aztec Diamonds, Tri Lai
A Simple Proof For The Number Of Tilings Of Quartered Aztec Diamonds, Tri Lai
Department of Mathematics: Faculty Publications
We get four quartered Aztec diamonds by dividing an Aztec diamond region by two zigzag cuts passing its center. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of quartered Aztec diamonds is given by simple product formulas. In this paper we present a simple proof for this result.