  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.
