The generating function of convex polyominoes: The resolution of a q-differential system

We give a `beautiful? ? though complex ? formula for the generating function Z of convex polyominoes, according to their area, width and height. Our method consists in solving a linear q-differential system of size three, which was derived two years ago by encoding convex polyominoes with the words of an algebraic language (Schutzenberger's methodology). Three other formulas had already been obtained for Z, but neither was entirely satisfying.

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