Characterizations of Flip-Accessibility for Domino Tilings of the Whole Plane

A Characterization of Flip-accessibility for Rhombus Tilings of the Whole Plane

Flip Invariance for Domino Tilings of Three-Dimensional Regions with Two Floors

We investigate tilings of cubiculated regions with two simply connected floors by 2×1×1 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, c...

Distances on rhombus tilings

The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180◦ a hexagon made of three rhombi). Motivated by the study of a quasicrystal growth model, we are here interested in better understanding how “tight” rhombus tiling spaces are flip-connected. We introduce a lower ...

On the connectivity of three-dimensional tilings

We consider domino tilings of three-dimensional cubiculated manifolds with or without boundary, including subsets of Euclidean space and threedimensional tori. In particular, we are interested in the connected components of the space of tilings of such regions under local moves. Building on the work of the third and fourth authors [19], we allow two possible local moves, the flip and trit. Thes...

Alternating sign matrices and domino tilings

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 anoth...