  We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated
by the idempotents of the semigroup $T(X,P)$, which consists of all
transformations of the finite set $X$ preserving a non-uniform partition $P$.
We also classify and enumerate the idempotent generating sets of this minimal
possible size. This extends results of the first two authors in the uniform
case.
