  Consider 2n points on the unit circle and a reference dissection D of the
convex hull of the odd points. The accordion complex of D is the simplicial
complex of subsets of pairwise noncrossing diagonals with even endpoints that
cross a connected set of diagonals of the dissection D. In particular, this
complex is an associahedron when D is a triangulation, and a Stokes complex
when D is a quadrangulation. We exhibit a bijection between the facets of the
accordion complex of D and some dual objects called the serpent nests of D,
settling in particular a conjecture stated by F.~Chapoton (2016) in the case of
Stokes complexes.
