Ribbon tilings and multidimensional height functions

Ribbon Tilings and Multidimensional Height Functions

We fix n and say a square in the two-dimensional grid indexed by (x, y) has color c if x+ y ≡ c (mod n). A ribbon tile of order-n is a connected polyomino containing exactly one square of each color. We show that the set of order-n ribbon tilings of a simply connected region R is in one-to-one correspondence with a set of height functions from the vertices of R to Zn satisfying certain differen...

Ribbon Tilings From Spherical Ones

The problem of classifying all tile-k-transitive tilings of the innnite 2-dimensional ribbon (and pinched-ribbon) is shown to be solvable by classifying certain tile-k-transitive tilings of the sphere, for all k 2 N. Complete results are listed for k 3.

Generalized Tilings with Height

Height Function on Domino Tilings

We first define the height function on a domino tiling (as done in [1]) and state some of its basic properties. We then revisit the coupling function and relate it to Green’s function, which allows us to conclude that the coupling function converges in the limit to an analytic function with a pole. Using this, we do a general second moment calculation via the proof found in [1] which writes the...

Counting Fixed-Height Tatami Tilings

A tatami tiling is an arrangement of 1 × 2 dominoes (or mats) in a rectangle with m rows and n columns, subject to the constraint that no four corners meet at a point. For fixed m we present and use Dean Hickerson’s combinatorial decomposition of the set of tatami tilings — a decomposition that allows them to be viewed as certain classes of restricted compositions when n ≥ m. Using this decompo...