Alternating sign matrices and tilings of Aztec rectangles
نویسنده
چکیده
The problem of counting numbers of tilings of certain regions has long interested researchers in a variety of disciplines. In recent years, many beautiful results have been obtained related to the enumeration of tilings of particular regions called Aztec diamonds. Problems currently under investigation include counting the tilings of related regions with holes and describing the behavior of random tilings. Here we derive a recurrence relation for the number of domino tilings of Aztec rectangles with squares removed along one or both of the long edges. Through an interpretation of a sequence of alternating sign matrix rows as a family of nonintersecting lattice paths, we relate this enumeration to that of lozenge tilings of trapezoids, and use the LindströmGessel-Viennot theorem to express the number in terms of determinants.
منابع مشابه
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 ...
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We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order n but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from anoth...
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