Enumerative properties of generalized associahedra

TL;DR

This paper explores the enumerative properties of generalized associahedra, connecting their combinatorial structures with lattice theory and spectral sequences, and proposes a conjecture relating them to noncrossing partitions.

Contribution

It introduces new enumerative relations for generalized associahedra and provides evidence supporting a conjecture linking them to noncrossing partition lattices.

Findings

Enumerative relations with generalized noncrossing partitions are proposed.

Evidence supports the conjectured connection between associahedra and noncrossing partitions.

Spectral sequences of a related bicomplex are used to analyze these properties.

Abstract

Some enumerative aspects of the fans, called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras are considered, in relation with a bicomplex and its two spectral sequences. A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given.