  Partition algebras with non-zero parameters are cellularly stratified and
thus have the features of both cellular algebras and stratified algebras. Also,
partition algebras form a tower of algebras. In this paper, we provide a
diagrammatic approach to study the branching rules for cell modules on a tower
of the partition algebras. This also allows us to calculate the structure
constants of the Grothendieck ring of the tower.
