A reciprocity theorem for domino tilings

A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted ${\mathcal T}_n$, are obtained by starting with a square of side-length $2n$, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of ord...

We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.

We consider tilings of deficient rectangles by the set $\mathcal{T}_4$ of ribbon $L$-tetrominoes. A tiling exists iff the rectangle is a square of odd side. The missing cell is on the main NW--SE diagonal, in an odd position if the square is $(4m+1)\times (4m+1)$ and in an even position for $(4m+3)\times (4m+3)$. The majority of the tiles in a tiling are paired and each pair tiles a $2\times 4$ rectangle. The tiles in an irregular position and the missing cell form a NW--SE d...

We look at sets of tiles that can tile any region of size greater than 1 on the square grid. This is not the typical tiling question, but relates closely to it and therefore can help solve other tiling problems -- we give an example of this. We also present a result to a more classic tiling question with dominoes and L-shape tiles.

Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an $(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two $(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and $\{C,D\}\neq \{A,B\}$, then the pair $C,D$ is called a...

In their unpublished work, Jockusch and Propp showed that a 2-enumeration of antisymmetric monotone triangles is given by a simple product formula. On the other hand, the author proved that the same formula counts the domino tilings of the quartered Aztec rectangle. In this paper, we explain this phenomenon directly by building a correspondence between the antisymmetric monotone triangles and domino tilings of the quartered Aztec rectangle.

An \emph{auspicious tatami mat arrangement} is a tiling of a rectilinear region with two types of tiles, $1 \times 2$ tiles (dimers) and $1 \times 1$ tiles (monomers). The tiles must cover the region and satisfy the constraint that no four corners of the tiles meet; such tilings are called \emph{tatami tilings}. The main focus of this paper is when the rectilinear region is a rectangle. We provide a structural characterization of rectangular tatami tilings and use it to prove...

Tilings are around us everywhere, and our curiosity draws us to study their properties. A tiling is a way of arranging pieces on a board, such that there is no space left uncovered, nor any space covered by more than one tile. In particular, we study fault-free tilings of boards with dominoes. To be fault-free every line that intersects the tiling must also intersect the interior of at least one of the tiles. Fault-free rectangular boards have been well studied, however we lo...

In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing ``barriers'' in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of constrained tilings is independent of the placement of the barriers. We do not know of a combinatorial explanation of this fact; our proof uses the Jacobi-Trudi identity.

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions. In earlier work, Stanley proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp, Stanley's result concerns th...