Alternating sign matrices and domino tilings

Duke Mathematical Journal, 1996

We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well.

Journal of Combinatorial Theory, Series B, 2005

The Aztec diamond of order n is a certain configuration of 2n(n + 1) unit squares. We give a new proof of the fact that the number Π n of tilings of the Aztec diamond of order n with dominoes equals 2 n(n+1)/2 . We determine a sign-nonsingular matrix of order n(n + 1) whose determinant gives Π n . We reduce the calculation of this determinant to that of a Hankel matrix of order n whose entries are large Schröder numbers. To calculate that determinant we make use of the J-fraction expansion of the generating function of the Schröder numbers.

Arxiv preprint math/9801068

In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/ √ 2 for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.

This paper reports a work in progress whose aim is to develop a computational framework towards the generation and analysis of rectangular domino tilings, that is, nonoverlapping coverings of a given rectangle by dominoes. A given rectangle is assumed to be of size n × m, n and m positive integers and all tessellating dominoes are identical rectangles of size 1 × 2. Our problem is to automatically obtain the generating function of the number of domino tilings the rectangle possess.

Jct, 2005

The Aztec diamond of order n is a certain configuration of 2n(n + 1) unit squares. We give a new proof of the fact that the number Π n of tilings of the Aztec diamond of order n with dominoes equals 2 n(n+1)/2 . We determine a sign-nonsingular matrix of order n(n + 1) whose determinant gives Π n . We reduce the calculation of this determinant to that of a Hankel matrix of order n whose entries are large Schröder numbers. To calculate that determinant we make use of the J-fraction expansion of the generating function of the Schröder numbers.

We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.

We discuss tilings of a grid (of size $n × 2$) with dominoes of size $2 × 1.$ Parameters that might be called " longest run " are investigated, in terms of generating functions and also asymptotically. Extensions are also mentioned.

Graphs and Combinatorics, 2015

The Aztec diamond of order n is the union of lattice squares in the plane intersecting the square |x| + |y| < n. The Aztec diamond theorem states that the number of domino tilings of this shape is 2 n(n+1)/2. It was first proved by Elkies, Kuperberg, Larsen and Propp in 1992. We give a new simple proof of this theorem.

Theoretical Computer Science, 2013

We study the combinatorial properties and the problem of generating exhaustively double square tiles, i.e. polyominoes yielding two distinct periodic tilings by translated copies such that every polyomino in the tiling is surrounded by exactly four copies. We show in particular that every prime double square tile may be obtained from the unit square by applying successively some invertible operators on double squares. As a consequence, we prove a conjecture of Provençal and Vuillon [17] stating that these polyominoes are invariant under rotation by angle π.

Theoretical Computer Science, 2004

We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called flips.