A reciprocity theorem for domino tilings

The tiling problem has been a famous problem that has appeared in many Mathematics problems. Many of its solutions are rooted in high-level Mathematics. Thus we hope to tackle this problem using more elementary Mathematics concepts. In this report, we start with the simplest cases, with the smaller numbers: the number of ways to tile a $2 \times n$, $3 \times n$, $4 \times n$ rectangular board using $2 \times 1$ domino tiles, where the number of rows is fixed and we present a...

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 x 2k rectangle is congruent to 1 mod 4.

We consider tromino tilings of $m\times n$ domino-deficient rectangles, where $3|(mn-2)$ and $m,n\geq0$, and characterize all cases of domino removal that admit such tilings, thereby settling the open problem posed by J. M. Ash and S. Golomb in \cite {marshall}. Based on this characterization, we design a procedure for constructing such a tiling if one exists. We also consider the problem of counting such tilings and derive the exact formula for the number of tilings for $2\t...

We study the 2-adic behavior of the number of domino tilings of a 2n-by-2n square as nvaries. It was previously known that this number was of the form 2^n f(n)^2, where f(n) is an odd, positive integer. We show that the function f is uniformly continuous under the 2-adic metric, and thus extends to a function on all of Z. The extension satisfies the functional equation f(-1-n) = +- f(n), where +- sign is + if n is congruent to 0 or 3 modulo 4 and - otherwise.

In this paper, we give inductive sum formulas to calculate the number of diagonally symmetric, and diagonally \& anti-diagonally symmetric domino tilings of Aztec Diamonds. As a byproduct, we also find such a formula for the unrestricted case as well. Our proofs rely on a new technique for counting the number of perfect matchings of graphs, proposed by the authors recently.

We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on the $(2k+1)\times (2k+1)$ square grid, we show that the number of near-perfect matchings is a multiple of $2^k$, and from this follows a conjecture of Kong that the total number of near-perfect matchings is a multiple of $2^k$. We also determ...

In this paper we study different kinds of symmetries related to the domino tilings of chessboards.

We prove combinatorially that the parity of the number of domino tilings of a region is equal to the parity of the number of domino tilings of a particular subregion. Using this result we can resolve the holey square conjecture. We furthermore give combinatorial proofs of several other tiling parity results, including that the number of domino tilings of a particular family of rectangles is always odd.

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 ...

In this paper a closed form expression for the number of tilings of an $n\times n$ square border with $1\times 1$ and $2\times1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $m\times n$ rectangular borders. The number of distinct tilings up to rotational symmetry is considered, and closed form expressions are given, in the case of a square border and in the case of a rectangular border. Finally, the number of distinct tilin...