Automatic CounTilings

Traditionally, a human would have had to tackle each specific width (k) and each specific set of tiles (T ), (even the set consisting of the two dimers, the vertical and horizontal 1 by 2 rectangles) one at a time. He or she would have had to figure out the combinatorics of the situation, establish a “structure theorem”, then deduce from this a set of equations, and finally, solve this set of equations (this very last part, starting in the seventies, was possibly aided by Maple or Mathematica). Of course, the human will only be able to do it for very small k, and very simple sets of tiles, since very soon the structure theorem, and consequently the set of equations, would be too hard to derive by hand, let alone solve.