On the H-triangle of generalised nonnesting partitions

To a crystallographic root system Φ, and a positive integer k, there are associated two Fuss-Catalan objects, the set of nonnesting partitions NN^(k)(Φ), and the cluster complex Δ^(k)(Φ). 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)(Φ) along the way, including an earlier conjecture by Fomin and Reading giving a refined enumeration by Fuß-Narayana numbers.

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