On Integer Partitions and Continued Fraction Type Algorithms

Abstract We show that the additive-slow-Farey version of the traditional continued fraction 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, in terms of the classical Farey tree. We give complete description of the conjugation for the corresponding Young shapes via the dynamics of the Farey tree. We then do the analog using the additive-slow-Farey version of the triangle map (a type of multi-dimensional continued fraction algorithm), giving us a a method for producing integer partitions of a positive number n into three smaller numbers, with multiplicity. In this case we also give complete description of the conjugation for the corresponding Young shapes via the dynamics of the triangle map. We then give a criterion that starts to determine which partitions of a positive number n into three smaller numbers, with multiplicity, can occur via the triangle map.