Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux

TL;DR

This paper explores the structure of triangle-free triangulations of convex polygons, linking diagonal flips to hyperplane arrangements and Young tableaux, revealing unique properties of geodesics and flip counts.

Contribution

It introduces a novel interpretation of diagonal flips as hyperplane arrangement moves and establishes new combinatorial relationships involving Young tableaux.

Findings

Each diagonal is flipped exactly once in a geodesic between antipodes

Number of geodesics equals twice the count of certain Young tableaux

Properties of geodesics in flip graphs are characterized

Abstract

Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between distinguished pairs of antipodes; (2) the number of geodesics between these antipodes is equal to twice the number of Young tableaux of a truncated shifted staircase shape.