  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.

Rings and Algebras;Combinatorics