  I present a bijection on integer partitions that leads to recursive
expressions, closed formulae and generating functions for the cardinality of
certain sets of partitions of a positive integer $n$. The bijection leads also
to a product on partitions that is associative with a natural grading thus
defining a free associative algebra on the set of integer partitions. As an
outcome of the computations, certain sets of integers appear that I call
difference sets and the product of the integers in a difference set is an
invariant for a family of sets of partitions. The main combinatorial objects
used in these constructions are the central hooks of the Ferrers diagrams of
partitions.
