## 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, that move by unit steps in any of the following directions: West,
North-East, East and South-West. 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. We propose in [3] 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. This work
has been published in the prestigious journal *Transactions of the AMS*.