Enumeration of idempotents in diagram semigroups and algebras

TL;DR

This paper characterizes and enumerates idempotents in various diagram semigroups and algebras, providing formulas, recursions, and class counts, advancing understanding of their algebraic structure.

Contribution

It offers a new characterization of idempotents in the partition monoid and derives enumeration formulas for idempotents in several related semigroups and algebras.

Findings

Derived formulas and recursions for counting idempotents.

Enumerated idempotents in partition, Brauer, and partial Brauer monoids.

Determined the number of idempotent basis elements in associated 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 $\mathscr R$-, $\mathscr L$- 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.