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Written: Jan. 20, 2006.
published in the
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
and copublished (July 2009) in
Rejecta Mathematica
This is yet another exercise in "symbol crunching".
What is so nice about it is that everything is done by machine:
the combinatorics, the algebra, and the analysis. Enjoy!
If I had Maple and this package back in 1960 I might have scooped
Kasteleyn and Fisher&Temperley.
Important: This article is accompanied by Maple
package
TILINGS
that automatically computes generating functions for the enumerating
sequences of tilings of fixedwidth rectangular boards with an arbitrary
set of tiles, and also does "statistical processing".
See also the precursor package,
RecTILINGS that only handles rectangular tiles.
Sample Input and Output

For the generating functions, first few terms, and asymptotic statistics for
the classical dimer problem (with the two domino tiles) for tiling rectangular
boards of width 2,4,6,8,
the input
yields the output.

For the generating functions, first few terms, and asymptotic statistics for
the monomerdimer problem (with the two domino tiles as well as a monomer (1x1) tile ) for tiling rectangular
boards of width 2,3,4,5,
the
input
yields the
output.

For the generating functions, first few terms, and asymptotic statistics for
tilings with the set of tiles consisting of a monomer (1x1), the two dimers
(1x2 and 2x1) and the 2x2 tiles, for tiling rectangular
boards of width 2,3,4
the
input
yields the
output.

For the generating functions, first few terms, and asymptotic growth of
tilings with the set of tiles consisting of a monomer (1x1), and the
1x3 and 3x1 tiles, for tiling rectangular
boards of width 1,2,3,4,5
the
input
yields the
output.

To prove Kasteleyn's famous formula (straight enumeration version) for
m=2,4,6,8,10, but ALL n,
the
input
yields the
output.
Added April 14, 2006: This article was submitted to Journal of Combinatorial
Theory Series A, and was stupidly rejected by
its Managing Editor, Helene Barcelo, and
the members of its Advisory Board, that includes, surprisingly and sadly,
Tiling expert Mireille BousquetMelou.
Read the
NarrowMinded and Ignorant Referee's Report and my response
.
Luckily, nowadays, publishing in a "real" journal is really only a formality,
and this masterpiece is hence published in this Personal Journal.
Added July 24, 2006: Read
Renzo Sprugnoli's interesting message
Personal Journal of Ekhad and Zeilberger
Doron Zeilberger's Home Page