| I Have a Dream Automatic CounTilings (2008) | |||||||||||||
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| One day it would be possible to write in English, or in an English-like super-high-level programming language, the following command: Write a Maple program that inputs an arbitrary positive integer k, a symbol n, and an arbitrary set T of tiles, as well as a symbol t, and outputs the rational function of t whose Maclaurin series’s coefficient of t n is the number of different tilings of a k by n rectangle by the tiles of T. But Yesterday’s Dream Came To Pass This day hasn’t arrived yet, so I had to spend a few weeks writing such a program myself. This is the human-generated Maple package TILINGS accompanying this article. But, this is already a great step forward. Once I wrote the program, my beloved servant, Shalosh B. Ekhad, can solve (potentially) infinitely many enumeration problems, completely rigorously, for enumerating tilings of rectangles of arbitrary width and using an arbitrary set of tiles (the tiles even do not have to be connected). 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 | |||||||||||||
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