Peeling Sequences

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 $12.{29}^n$ .

Keywords: Convexity; Integer sequence; Recursive construction.

Fig. 1

Illustration for $n=4,5,6$

Fig. 2

The first construction: before and after the flattening step

Fig. 3

The second construction