On integer partitions and continued fraction type algorithms

Our goal is to show that the additive-slow-Farey version of the Triangle map (a type of multi-dimensional continued fraction algorithm) gives us a method for producing a map from the set of integer partitions of a positive number n into itself. We start by showing 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 turn to the general case, using the the Triangle map to give a natural map from general integer partitions of a positive number n to integer partitions of n. This map will still be compatible with conjugation of the corresponding Young shapes. We will close by the observation that it appears few other multi-dimensional continued fraction algorithms can be used to study partitions.