A new perspective on k-triangulations

TL;DR

This paper links k-triangulations of convex polygons to Schubert polynomial theory, proving their complex is a vertex-decomposable sphere and providing a new proof for their enumeration formula.

Contribution

It introduces a novel connection between k-triangulations and Schubert polynomials, establishing topological and enumerative properties.

Findings

The simplicial complex of k-triangulations is a vertex-decomposable sphere.

A new proof of the determinantal formula for counting k-triangulations.

Established a link between geometric combinatorics and algebraic geometry.

Abstract

We connect k-triangulations of a convex n-gon to the theory of Schubert polynomials. We use this connection to prove that the simplicial complex with k-triangulations as facets is a vertex-decomposable triangulated sphere, and we give a new proof of the determinantal formula for the number of k-triangulations.