MacMahon Partition Analysis: A discrete approach to broken stick problems

TL;DR

This paper introduces a discrete method using MacMahon's Partition Analysis to solve polygon formation problems from broken sticks, providing a new way to compute probabilities involving integer-sided polygons and Diophantine inequalities.

Contribution

It applies MacMahon's Partition Analysis to derive generating functions for broken stick problems, enabling the calculation of complex probabilities that were previously difficult to generalize.

Findings

Derived a generating function for polygon formation probabilities

Established a formula involving generalized Fibonacci numbers

Solved a longstanding problem in broken stick probability calculations

Abstract

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for $n\geq k\geq 3$, the probability that a $k$-gon cannot be formed from a stick broken into $n$ parts is given by $n!$ over a product of linear combinations of partial sums of generalized Fibonacci numbers, a problem which proved to be very hard to generalize in the past.