Enumeration of idempotents in diagram semigroups and algebras

Abstract

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 $\mathcal{R}$-, $\mathcal{L}$- and $\mathcal{D}$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.

Small changes to intro and references. To appear in Journal of Combinatorial Theory, Series A. 26 pages, 4 figures, 10 tables

References in corpus (3)

• Free idempotent generated semigroups and endomorphism monoids of free $G$-acts
• Infinite partition monoids
• The Finite Basis Problem for Kiselman Monoids

Cited by in corpus (7)

• Motzkin monoids and partial Brauer monoids
• Idempotent generation in the endomorphism monoid of a uniform partition
• Sandwich semigroups in diagram categories
• Twisted Brauer monoids
• Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logic
• Identities of the Kauffman Monoid ${\mathcal{K}}_3$
• Classification of variants of partial Brauer monoids