Alternating-Sign Matrices and Domino Tilings (Part I)
نویسنده
چکیده
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 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.
منابع مشابه
Domino Tilings of Aztec Diamonds, Baxter Permutations, and Snow Leopard Permutations∗
In 1992 Elkies, Kuperberg, Larsen, and Propp introduced a bijection between domino tilings of Aztec diamonds and certain pairs of alternating-sign matrices whose sizes differ by one. In this paper we first study those smaller permutations which, when viewed as matrices, are paired with the matrices for doubly alternating Baxter permutations. We call these permutations snow ∗2010 AMS Subject Cla...
متن کاملAlternating sign matrices and tilings of Aztec rectangles
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 ran...
متن کاملAlternating sign matrices and domino tilings
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...
متن کاملAlternating-Sign Matrices and Domino Tilings (Part II)
We continue the study of the family of planar regions dubbed Aztec diamonds in our earlier article and study the ways in which these regions can be tiled by dominoes. Two more proofs of the main formula are given. The first uses the representation theory of GL(n). The second is more combinatorial and produces a generating function that gives not only the number of domino tilings of the Aztec di...
متن کاملBoundary correlation functions of the six - vertex model
We consider the six-vertex model on an N × N square lattice with the domain wall boundary conditions. Boundary one-point correlation functions of the model are expressed as determinants of N × N matrices, generalizing the known result for the partition function. In the free fermion case the explicit answers are obtained. The introduced correlation functions are closely related to the problem of...
متن کامل