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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, tilling 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 and will finish by introducing a recent algebraic approach with operators and commutations.
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