A conjecture presented in 1985 - the Andrews and Robbins conjecture - has recently been proved for the first time. It is thus clear that the structure which goes by the name of "totally symmetric plane partitions" can be described using a single formula. Producing the proof required vast computer resources and was only possible after the formula had been prepared for computer-assisted calculation.
This finding by a Austrian Science Fund FWF supported research group based in Linz, Austria will be published in the Proceedings of the National Academy of Sciences today. The proof means that the last of a long list of famous mathematical conjectures relating to plane partitions has finally been proved.
Even mathematicians play with building blocks. At least if they are interested in so-called plane partitions, which are visualized with columns of "building blocks" (cubes) on a surface resembling a chessboard. When "building" such plane partitions, the mathematicians must adhere to certain rules: No column may be higher than the width of the surface, or than another column behind it or left of it. The question of how many column permutations may be built on a given surface area is easily answered, thanks to a specific formula. However, it becomes trickier if the permutations must follow stipulated symmetries, or if, instead of counting the permutations, you wish to count its constituents. Although formulas have been designed to do this too, the crux of the matter is that not all of these formulas have really been proved to be accurate. It is only conjectured.
Although it may sound easy, it represents a great challenge, according to Dr. Koutschan: "This method does not work for every equation. The most important step was for us to convert the Andrews-Robbins conjecture into a suitable form for the computer to be able to prove it." The fact that the adjoint equation was really somewhat more complex than "U=V", is illustrated by its size: if it were printed, it would cover approximately 1 million A4 pages, which makes it probably the longest equation ever used in a mathematical proof.STANLEY`S LIST
Granted, such successful results are still an exception. However, this FWF project underscores the potential of computer-based proof. Given the great pace at which computer performance is advancing, such methods will perhaps one day even offer answers to the great unsolved questions in mathematics.Image and text will be available from Tuesday, 25th January 2011, 9 am CET onwards:
Original publication: A proof of George Andrews` and David Robbins` q-TSPP conjecture. C. Koutschan, M. Kauers, D. Zeilberger. DOI: 10.1073/pnas.1019186108Scientific contact:
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