Peeling Sequences

TL;DR

This paper investigates the number of possible sequences for removing points from a set in the plane, showing that the count varies significantly depending on the configuration, with bounds established for the minimum and maximum cases.

Contribution

It provides bounds on the number of peeling sequences for point sets, revealing the range of possible removal orders based on geometric configurations.

Findings

Maximum number of sequences is n! for convex position.

Minimum number of sequences is roughly 3^n.

Number of sequences can be as high as 12.29^n for certain configurations.

Abstract

Given a set of $n$ labeled points in general position in the plane, we remove all of its points one by one. At each step, one point from the convex hull of the remaining set is erased. In how many ways can the process be carried out? The answer obviously depends on the point set. If the points are in convex position, there are exactly $n!$ ways, which is the maximum number of ways for $n$ points. But what is the minimum number? It is shown that this number is (roughly) at least $3^n$ and at most $12.29^n$.