14h00: *The complete generating function for Gessel walks is algebraic,* Alin Bostan, Équipe-projet Algorithms.

The aim of the talk is to show how a difficult combinatorial problem has been recently solved using an experimental-mathematics approach combined with (rather involved) computer algebra techniques. More precisely, let *g*(*n*,*i*,*j*) denote the number of lattice walks in the quarter plane which start at the origin, end at the point (*i*,*j*), and consist of *n* unit steps going either west, south-west, east, or north-east. In the early nineties, Ira Gessel conjectured that the sequence of excursions *g*(*n*,0,0) is holonomic. We will present the computer-driven discovery and proof of the following generalization, obtained in August 2008 together with Manuel Kauers: the trivariate generating series of the sequence (*g*(*n*,*i*,*j*))_{n,i,j} is an algebraic function.

Virginie Collette Last modified: Mon Mar 30 14:53:58 CEST 2009