Alternating-Sign Matrices and Domino Tilings (Part I) |
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Authors: | Noam Elkies Greg Kuperberg Michael Larsen James Propp |
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Institution: | (1) Harvard University, Cambridge, MA, 02138;(2) University of California at Berkeley, Berkeley, CA, 94720;(3) University of Pennsylvania, Philadelphia, PA, 19104;(4) Massachusetts Institute of Technology, Cambridge, MA, 02139 |
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Abstract: | 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
n(n+1)/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 Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matrices; by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond. |
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Keywords: | tiling domino alternating-sign matrix monotone triangle representation square ice |
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