A human proof of Gessel’s lattice path conjecture
نویسندگان
چکیده
Gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either West, East, South-West or North-East. In 2001, Ira Gessel conjectured a nice closed-form expression for the number of Gessel walks ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, again using computer algebra tools, that the complete generating function of Gessel walks is algebraic. In this article we propose the first “human proofs” of these results. They are derived from a new expression for the generating function of Gessel walks in terms of Weierstrass zeta functions.
منابع مشابه
Proof of Ira Gessel’s Lattice Path Conjecture
We present a computer-aided, yet fully rigorous, proof of Ira Gessel’s tantalizingly simply-stated conjecture that the number of ways of walking 2n steps in the region x + y ≥ 0, y ≥ 0 of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals 16n (5/6)n(1/2)n (5/3)n(2)n .
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