A combinatorial version of Sylvester's four-point problem

TL;DR

This paper generalizes Sylvester's four-point problem to a combinatorial setting involving symmetric group expressions, conjecturing a probability of 1/4 for the analogous event.

Contribution

It introduces a combinatorial framework for Sylvester's problem and proposes a conjecture for the probability in this new context.

Findings

Conjecture that the probability is 1/4 in the generalized problem.

Uses Goodman and Pollack's classification to extend Sylvester's problem.

Provides a new perspective linking geometry and algebraic combinatorics.

Abstract

J. J. Sylvester's four-point problem asks for the probability that four points chosen uniformly at random in the plane have a triangle as their convex hull. Using a combinatorial classification of points in the plane due to Goodman and Pollack, we generalize Sylvester's problem to one involving reduced expressions for the long word in the symmetric group. We conjecture an answer of 1/4 for this new version of the problem.