  We give a characterisation of the idempotents of the partition monoid, and
use this to enumerate the idempotents in the finite partition, Brauer and
partial Brauer monoids, giving several formulae and recursions for the number
of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$-
and $\mathscr D$-classes. We also apply our results to determine the number of
idempotent basis elements in the finite dimensional partition, Brauer and
partial Brauer algebras.
