On the -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 N N ( 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 N N ( k ) ( Φ ) along the way, including an earlier conjecture by Fomin and Reading giving a refined enumeration by Fuß-Narayana numbers.