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 \(\mathscr R\)-, \(\math...

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Bibliographic Details
Published in:arXiv.org 2014-11
Main Authors: Dolinka, Igor, East, James, Evangelou, Athanasios, FitzGerald, Des, Ham, Nicholas, Hyde, James, Loughlin, Nicholas
Format: Article
Language:English
Subjects:
Enumeration
Monoids
Partitions
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1408.2021