Enumeration of basic ideals in type B

TL;DR

This paper counts specific algebraic structures called basic ideals in type B Lie algebras and their affine extensions, using combinatorial methods involving generalized Dyck-paths, resulting in explicit formulas and a new integer sequence.

Contribution

It introduces a combinatorial approach to enumerate basic ideals in affine type B Lie algebras, providing explicit formulas and discovering a new integer sequence.

Findings

Explicit formulas for counting basic ideals in affine type B Lie algebras

Introduction of a new integer sequence related to these ideals

Application of generalized Dyck-paths in algebraic enumeration

Abstract

The number of ad-nilpotent ideals of the Borel subalgebra of the classical Lie algebra of type B_n is determined using combinatorial arguments involving a generalization of Dyck-paths. We also solve a similar problem for the untwisted affine Lie algebra of type ~B_n, where we instead enumerate a certain class of ideals called basic ideals. This leads to an explicit formula for the number of basic ideals in ~B_n, which gives rise to a new integer sequence.