Enumeration of idempotents in planar diagram monoids

TL;DR

This paper classifies and counts idempotent elements in specific planar diagram monoids using graph-based algorithms, providing new enumeration methods and detailed tables of results.

Contribution

It introduces a novel graph-based classification and enumeration approach for idempotents in Motzkin, Jones, and Kauffman monoids, with algorithmic implementations and comparative analysis.

Findings

Developed algorithms for enumerating idempotents in planar diagram monoids.

Provided tables of calculated idempotent counts for various monoids.

Compared new algorithms with existing methods for semigroup idempotent enumeration.

Abstract

We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.