Published on *www.math.tifr.res.in* (http://www.math.tifr.res.in)

University of Bordeaux, France

February 25, 2010

At the beginning of this century, the physicists Razumov and Stroganov discovered a certain sequence of integers appearing in the study of some model for `quantum spin chains'. This sequence was already known by combinatorists in the enumeration of various classes of combinatorial objects: alternating sign matrices, 3D partitions of integers, tiling of hexagons on a triangular lattice. In the last 30 years, intensive studies have been made about these objects, beautiful and simple conjectured enumeration formulae had to wait for a long time before being proved. But many researches remain to be done in order to `understand' these formulae and the relationship with quantum spin chains model in physics.

No prerequisites are needed for this colloquium, neither in physics nor in combinatorics. I will give a short introduction to enumerative combinatorics an