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 R -, L - and 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. Enumeration of idempotents in diagram semigroups and algebras......Page 1 1 Introduction......Page 2 2 Preliminaries......Page 5 3 Characterisation of idempotents......Page 8 4 Enumeration of idempotents......Page 9 4.1 The partition monoid......Page 13 4.2 The Brauer monoid......Page 15 4.3 The partial Brauer monoid......Page 18 4.4 Other subsemigroups......Page 20 5 A different approach for Bn and PBn......Page 21 5.1 The Brauer monoid......Page 23 5.2 The partial Brauer monoid......Page 24 6 Idempotents in diagram algebras......Page 26 7 Calculated values......Page 31 References......Page 33