On the H-triangle of generalised nonnesting partitions

Abstract: To a crystallographic root system \Phi, and a positive integer k, there are associated two Fuss-Catalan objects, the set of nonnesting partitions NN^k(\Phi), and the cluster complex \Delta^k(\Phi). These posess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton for k=1 and later generalized to k>1 by Armstrong. We prove this conjecture, obtaining some structural and enumerative results on NN^k(\Phi) along the way, including an earlier conjecture by Fomin and Reading giving a refined enumeration by Fu{\ss}-Narayana numbers.