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Chapuy and Stump have given a nice generating series for the number offactorisations of a Coxeter element as a product of reflections. Their methodis to evaluate case by case a character-theoretic expression. The goal of thisnote is to give a uniform evaluation of their character-theoretic expression inthe case of Weyl groups, by using combinatorial properties of Deligne-Lusztigrepresentations.

"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element