We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

MSC Classifications:

20F55, 17B20, 05E99 show english descriptions Reflection and Coxeter groups [See also 22E40, 51F15]
Simple, semisimple, reductive (super)algebras
None of the above, but in this section 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]
17B20 - Simple, semisimple, reductive (super)algebras
05E99 - None of the above, but in this section

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