Geometry and complexity of O'Hara's algorithm

TL;DR

This paper investigates O'Hara's partition bijection, revealing its geometric interpretation, analyzing its computational complexity, and demonstrating efficient algorithms for specific cases, including finite support identities.

Contribution

It provides a geometric perspective, new complexity bounds, and polynomial-time algorithms for finite support cases of O'Hara's bijection.

Findings

Geometric interpretation as scissor congruence

Complexity bounds vary from efficient to mildly exponential

Polynomial-time computation for finite support identities

Abstract

In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general. Finally, we prove that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.