On integer partitions and continued fraction type algorithms

TL;DR

This paper links continued fraction algorithms with integer partitions, providing new formulas and interpretations for partition counts using Farey tree dynamics and multi-dimensional continued fractions.

Contribution

It introduces a novel interpretation of the additive-slow-Farey continued fractions as a method for generating integer partitions and derives new formulas for partition counts using Farey map dynamics.

Findings

Derived a new formula for p(2,n) using Farey map dynamics.

Established a method for generating partitions into three parts via the Triangle map.

Connected continued fractions with Young shape conjugation and partition enumeration.

Abstract

We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number $n$ into two smaller numbers, with multiplicity. We provide a complete description of how such integer partitions occur and of the conjugation for the corresponding Young shapes via the dynamics of the classical Farey tree. We use the dynamics of the Farey map to get a new formula for $p(2,n)$, the number of ways for partitioning $n$ into two smaller positive integers, with multiplicity. We then do the analogue using the additive-slow-Farey version of the Triangle map (a type of multi-dimensional continued fraction algorithm), giving us a method for producing integer partitions of a positive number $n$ into three smaller numbers, with multiplicity. However different aspects of this generalisations remain unclear.