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Alternating sign matrices and domino tilings

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arxiv math/9201305 v1 pith:LBKKSD57 submitted 1991-06-01 math.CO

Noam Elkies (Harvard) , Greg Kuperberg (UC Berkeley) , Michael Larsen (U Penn) , James Propp (MIT) This is my paper · ORCID
classification math.CO
keywords alternatingaztecdominodominoesmatricesregionssigntilings
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abstract

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 another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.

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  1. Frozen-corner enumeration of Alternating Sign Matrices

    math.CO 2025-09 conditional novelty 3.0 of 10

    The number of ASMs with an s by s frozen zero corner is conjectured to equal A_n det(1-M), a determinant formula verified numerically for all n up to 20.

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