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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).

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# Proof of Ira Gessel's Lattice Path Conjecture

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