Special Configurations in Anchored Rectangle Packings

TL;DR

This paper investigates anchored rectangle packings within the unit square, verifying a longstanding conjecture for specific point configurations based on permutation classes.

Contribution

It confirms Freedman's conjecture for certain classes of point configurations characterized by permutation properties.

Findings

Confirmed Freedman's conjecture for specific permutation classes.

Identified conditions under which anchored rectangle packings reach at least half the unit square.

Extended understanding of geometric packing problems with permutation-based configurations.

Abstract

Given a finite set S in $[0,1]^2$ including the origin, an anchored rectangle packing is a set of non-overlapping rectangles in the unit square where each rectangle has a point of S as its left-bottom corner and contains no point of S in its interior. Allen Freedman conjectured in the 1960's one can always find an anchored rectangle packing with total area at least $1/2$. We verify the conjecture for point configurations whose relative positions belong to certain classes of permutations.