Alternating-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 diamond of order n but also information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Lastly, we explore a connection between the combinatorial objects studied in this paper and the square-ice model studied by Lieb.     

Double Aztec rectangles

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies et al. (1992) [2], [3]). Moreover, we consider the connection between the generating function and MacMahon's q-enumeration of plane partitions fitting in a given box     

Domino tilings of the expanded Aztec diamond

The expanded Aztec diamond is a generalized version of the Aztec diamond, with an arbitrary number of long columns and long rows in the middle. In this paper, we count the number of domino tilings of the expanded Aztec diamond. The exact number of domino tilings is given by recurrence relations of state matrices by virtue of the state matrix recursion algorithm, recently developed by the author to solve various two-dimensional regular lattice model enumeration problems.     

A New Simple Proof of the Aztec Diamond Theorem

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 et al. (J. Algebraic Comb. 1(2):111–132, 1992). We give a new simple proof of this theorem.     

Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

A Toeplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Johansson.     

Domino tilings of Aztec diamonds and squares

An Aztec diamond of rank n is a rhombus of side length n on the square grid. We give several new combinatorial proofs of known theorems about the numbers of domino tilings of diamonds and squares. We also prove generalizations of these theorems for the generating polynomials of some statistics of tilings. Some results here are new. For example, we describe how to calculate the rank of a tiling using special weights of edges on the square grid. Bibliography: 17 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 180–230.     

Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid G _n of order n is similar to the disjoint union of two copies of the quartered Aztec diamond QAD _n−1 of order n−1 with the path P _n ⁽²⁾ on n vertices having edge weights equal to 2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. It follows in particular that the characteristic polynomials of the above graphs satisfy the equality P(G _n )=P(P _n ⁽²⁾)[P(QAD _n−1)]². On the one hand, this provides a combinatorial explanation for the “squarishness” of the characteristic polynomial of the square grid—i.e., that it is a perfect square, up to a factor of relatively small degree. On the other hand, as formulas for the characteristic polynomials of the path and the square grid are well known, our equality determines the characteristic polynomial of the quartered Aztec diamond. In turn, the latter allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described “linear squarishness” property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases—graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases. Research supported in part by NSF grant DMS-0500616.     

Generating Function of the Tilings of an Aztec Rectangle with Holes

We consider a generating function of the domino tilings of an Aztec rectangle with several unit squares removed from the boundary. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.     

Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

Computing a pyramid partition generating function with dimer shuffling

We verify a recent conjecture of Kenyon/Szendr?i by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁x₂−x₃x₄=0}⊂C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp [Noam Elkies, Greg Kuperberg, Michael Larsen, James Propp, Alternating sign matrices and domino tilings. II, J. Algebraic Combin. 1 (3) (1992) 219-234].     

A visual proof of a result of Knuth on spanning trees of Aztec diamonds in the case of odd order

The even Aztec diamond AD_n is known to have precisely four times more spanning trees than the odd Aztec diamond OD_n—this was conjectured by Stanley and first proved by Knuth. We present a short combinatorial proof of this fact in the case of odd n. Our proof works also for the more general case of odd-by-odd Aztec rectangles.     

Double Aztec diamonds and the tacnode process

Discrete and continuous non-intersecting random processes have given rise to critical “infinite-dimensional diffusions”, like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary.     

Singular Polynomials of Generalized Kasteleyn Matrices

Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK* (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles.     

A new proof of the refined alternating sign matrix theorem

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n×n alternating sign matrices with a 1 at the top of the ith column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(−1) matrices generalizing alternating sign matrices.     

Coupling functions for domino tilings of Aztec diamonds

The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, that we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q^vol$q^{\mathrm{vol}}$ weighting which corresponds to multiplying the product of tile weights by q if we add a ‘box’ to the height function and the two-periodic weighting which exhibits a flat region with defects in the center.     

State matrix recursion method and monomer–dimer problem

The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.     

An arctic circle theorem for Groves

In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable ‘temperate zone’ in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds.Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds.     

Perfect Matchings and Perfect Powers

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin. 1 (1992), 111–132; B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991; J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfect matchings of a certain family of subgraphs of the square lattice is a power of 3 or twice a power of 3. In addition we obtain multi-parameter generalizations of previously known results, and new multi-parameter exact enumeration results. We obtain in particular a simple combinatorial proof of Bo-Yin Yang's multivariate generalization of fortresses, a result whose previously known proof was quite complicated, amounting to evaluation of the Kasteleyn matrix by explicit row reduction. We also include a new multivariate exact enumeration of Aztec diamonds, in the spirit of Stanley's multivariate version.     

Non-intersecting paths, random tilings and random matrices

 We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from non-intersecting Brownian motions. The derivations of the measures are based on the Karlin-McGregor or Lindstr?m-Gessel-Viennot method. We use the measures to show some asymptotic results for the models. Received: 1 December 2000 / Revised version: 20 May 2001 / Published online: 17 May 2002     

Domino Tiling Congruence Modulo 4

The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k × 2k rectangle is congruent to 1 mod 4.