I'd like to see (or figure out) a proof of that determinant-count principle. Also, I recall vaguely that on math-fun ca. 20 years ago, Bill Thurston was talking about some clever insight he had into domino tilings of plane regions. I wonder if someone can look into math-fun archives and re-post his comments. —Dan
On Saturday/26December/2020, at 6:43 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Today I learned from a Mathologer video (Burkard Polster, who apparently knows funster Jim Propp), that there is a fairly simple formula for the number of domino tilings on an arbitrary simply-connected polyomino. (Mathologer was ambiguous about whether the formula fails when the polyomino has narrow "necks", but my guess is that these cases are fine.)
You checkerboard-color the thing. In order for there to be any tilings at all, there have to be an equal number n of black and white squares. Then, assigning columns to white and rows to black, you construct an adjacency matrix; horizontal connections are marked 1, vertical connections are marked i (!), with 0's elsewhere; and then finally you take the determinant. Magically, either the entire real part or the entire imaginary part cancels away, and the absolute coefficient of the result (either pure imaginary or pure real) is the number of tilings.