Proof of Ira Gessel's Lattice Path Conjecture

[Count Iblis]

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gessel.html"

In a recent article, Manuel Kauers and I tried very hard to prove Ira Gessel's notorious conjecture, that has been circulating in combinatorial enumeration circles for the last seven years, about the number of ways of walking, in the "Manhattan lattice" (2D square-lattice), 2n steps, from the origin back to the origin, using unit steps in the four fundamental directions (north, south, east, and west), all the while staying in x+y ≥ 0, y ≥ 0. Ira Gessel conjectured that it is given by the beautiful expression

[ 16^n (5/6)_ n (1/2)_n]/[(5/3)_n (2)_n] ,

where (a)_n=a(a+1)...(a+n-1) .

We failed, becuase our computers ran out of memory, even though we felt that a sufficiently large computer would yield to our approach. But then came along the brilliant Christoph Koutschan, and joined the effort, and together with Manuel, was able to complete the task, still using our ideas, but adding to them some very good ones of his own, and this lead to the final solution.


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Important: This article is accompanied by the Maple file Guessel2 that has the annihilating operator described in the paper, and that verifies that Gessel's expression does indeed satisfy it. (in Maple, type bdok1(n); and see whether you get 0).

 

[mmwave]

Dear fellow scientists,

I am very impressed by your dedication and persistence in attempting to solve Ira Gessel's conjecture. Your collaboration with Christoph Koutschan and use of advanced technology such as the Maple file Guessel2 is truly commendable. It is clear that you have put a lot of effort and thought into this problem, and your results are truly remarkable.

I understand the frustration of encountering limitations with our computational resources, but it is inspiring to see that you did not give up and found a way to overcome this obstacle. Your approach of combining your own ideas with those of others is a great example of the power of collaboration in scientific research.

I would also like to thank you for sharing the Maple file Guessel2 with us. This will be a valuable resource for other researchers who are interested in this problem. Your verification of Gessel's expression using this file is a strong indication of the validity of your solution.

Overall, I am very impressed by your work and I am sure that your solution will have a significant impact in the field of combinatorial enumeration. Thank you for your dedication and for pushing the boundaries of scientific knowledge.