Cubicality of subdivisions of type A generalized associahedra

Prove that the analogous subdivisions of all type A generalized associahedra are composed of exactly $(n-1)!$ cubes.

Background

The paper constructs a cubical subdivision of the Loday realization of the associahedron and then places this construction in the broader framework of type A generalized associahedra. The authors state that analogous subdivisions can be obtained for all type A generalized associahedra, but explicitly leave unresolved whether every such subdivision consists of (n−1)!(n-1)! cubes. They also indicate that the combinatorics of these cases are not studied in detail.

References

We obtain similar subdivisions of all type A generalized associahedra which we conjecture to also all be composed of $(n-1)!$ cubes, but we do not undertake a detailed study of the combinatorics in these cases.

— A cubulation of the associahedron using a torus-equivariant degeneration of the brick variety  (2610.23004 - Gandhi et al., 16 Oct 2026) in Introduction, paragraph discussing generalized associahedra

We conjecture this construction yields a dissection of the faces of the associahedron which has a face lattice isomorphic to that of the permutahedron.

— A cubulation of the associahedron using a torus-equivariant degeneration of the brick variety  (2610.23004 - Gandhi et al., 16 Oct 2026) in Section 1, subsection “Cubulating by thinking inside of the box”