This talk will link partition numbers from combinatorics with a certain multi-dimensional continued fraction algorithm (the triangle map) from number theory and dynamical systems.

Andrew and Eriksson’s Introduction to Integer Partitions starts with discussing Euler’s identity: “Every number has as many integer partitions into odd parts as into distinct parts.” As they state, this is quite surprising if you have never seen it before. There are, though, many other equally if not more surprising partition identities. For all there are two basic questions. First, how to even guess the existence of any potential partition identities. Then, once a possible potential identity is conjectured, how to prove it.

In joint work with Baalbaki, Bonanno, Del Vigna and Isola, there was developed a link between traditional continued fractions and the slow triangle map (a type of multi-dimensional continued fraction algorithm) with integer partitions. These maps were initially introduced for number theoretic reasons but have over the years exhibited many interesting dynamical properties. We will see that these maps create a natural map (an almost internal symmetry) from the set of integer partitions to itself.

Thus we will allow us to create a new technique for generating any number of partition identities.

We will start with an introduction to the triangle map (and multi-dimensional continued fractions in general), and then see how it is linked to partition numbers.