# The serpent nest conjecture for accordion complexes

## TL;DR

This paper introduces the accordion complex for certain polygon dissections, establishes a bijection with serpent nests, and confirms a conjecture related to Stokes complexes, expanding understanding of these combinatorial structures.

## Contribution

It proves a conjecture by F. Chapoton by establishing a bijection between facets of the accordion complex and serpent nests for Stokes complexes.

## Key findings

- Bijection between facets of the accordion complex and serpent nests.
- Confirmation of Chapoton's conjecture for Stokes complexes.
- Extension of associahedron and Stokes complex concepts.

## Abstract

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.