Enumeration of idempotents in diagram semigroups and algebras
  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. 
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald,   Nicholas Ham, James Hyde, Nicholas Loughlin
math.GR math.CO math.RT