Stokes posets and serpent nests

TL;DR

This paper explores two combinatorial structures linked to quadrangulations of polygons, generalizing cluster algebra and nonnesting partition concepts, with potential implications for algebraic and geometric combinatorics.

Contribution

It introduces new posets and path configurations associated with quadrangulations, extending known combinatorial frameworks in cluster algebras.

Findings

Defined a new poset related to Stokes polytopes

Constructed specific path configurations within quadrangulations

Generalized combinatorics of cluster algebras and nonnesting partitions

Abstract

We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations inside the quadrangulation, satisfying some specific constraints. These objects provide a generalisation of the existing combinatorics of cluster algebras and nonnesting partitions of type A.