Abstract
With a crystallographic root system $\Phi$ , there are associated two Catalan objects, the set of nonnesting partitions $NN(\Phi)$, and the cluster complex $\Delta (\Phi)$. These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton. We prove this conjecture, and indicate its generalisation for the Fuß-Catalan objects $NN^{(k)}(\Phi)$ and $\Delta^{(k)}(\Phi)$, conjectured by Armstrong. A un système de racines cristallographique, on associe deux objets de Catalan: l’ensemble des partitions non-emboîtées $NN(\Phi)$, et le complexe d’amas$\Delta (\Phi)$. Ils possèdent de nombreuses coïncidences énumératives, plusieurs d’entre elles étant capturées dans une identité surprenante, conjecturée par Chapoton. Nous démontrons cette conjecture, et indiquons sa généralisation pour les objets de Fuß-Catalan $NN^{(k)}(\Phi)$ et $\Delta^{(k)}(\Phi)$, conjecturée par Armstrong.
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