Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Ron M. Adin, Yuval Roichman

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
