On the H-triangle of generalised nonnesting partitions

TL;DR

This paper proves a conjecture linking two Fuss-Catalan objects associated with crystallographic root systems, revealing structural and enumerative properties of generalized nonnesting partitions and cluster complexes.

Contribution

It proves Chapoton's conjecture for all positive integers k, establishing a deep connection between nonnesting partitions and cluster complexes for crystallographic root systems.

Findings

Proved the identity conjectured by Chapoton for all k.

Derived structural results on NN^(k)() and cluster complexes.

Provided refined enumeration using Fuss-Narayana numbers.

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.