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In 2001 Razumov and Stroganov conjectured that the (properly normalized) components of the ground state of the dense O(1) loop model on a semi-infinite cylinder enumerate fully-packed loop (FPL) configurations on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points. This conjecture has arisen a lot of interest both in the physics and in the mathematics community. In this talk, after reviewing the main background, I shall discuss a stronger (but easier to prove!) version of the correspondence: on the `dense O(1) side', the ground state of Hamiltonian H of the loop model is replaced by the one of the Scattering Matrix S(t); on the `FPL side', one considers the refinement according to the position of the unique straight tile on the last row.

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