On the H-triangle of generalised nonnesting partitions

With a crystallographic root system Φ and a positive integer  k, there are associated two Fuß-Catalan objects, the set of k-generalised nonnesting partitions NN(k)(Φ), and the generalised cluster complex Δ(k)(Φ). These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton for k=1 and later generalised to k≥1 by Armstrong. We prove this conjecture, obtaining some structural and enumerative results on NN(k)(Φ) along the way, including an earlier conjecture by Fomin and Reading giving a refined enumeration by Fuß-Narayana numbers.