## Section: New Results

### A Human Proof of the Gessel Conjecture

Counting lattice paths obeying various geometric constraints is a classical
topic in combinatorics and probability theory. Many recent works deal with the
enumeration of 2-dimensional walks with prescribed steps confined to the
positive quadrant. A notoriously difficult case concerns the so-called
*Gessel walks*: they are planar walks confined to the positive quarter
plane, which move by unit steps in any of the West,
North-East, East, and South-West directions.
In 2001, Ira Gessel conjectured a closed-form
expression for the number of such walks of a given length starting and 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, using again
computer algebra tools, that the trivariate generating function of Gessel
walks is algebraic.
This year, Alin Bostan, together with Irina Kurkova (Univ. Paris 6)
and Kilian Raschel (CNRS and Univ. Tours), proposed
in [6] the first “human proofs” of these results.
They are derived from a new expression for the generating function of Gessel
walks in terms of special functions.