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Enumeration in the Chevalley algebras

We consider the maximal nilpotent subalgebra NΦ(<i>K</i>) of the Chevalley algebra of the type Φ over an arbitrary field <i>K</i>. Our purpose is enumeration of ideals of NΦ(<i>K</i>) that are invariant under the subgroup <i>D</i> of all diagonal automorphisms. Connections between ideals of NΦ(<i>K</i>) and normal subgroups of the unipotent subgroup of the Chevalley group of the type Φ over <i>K</i> are used. Finally, we use methods of integral representation of sums and of listing lattice paths. We also discuss some other problems of the enumeration of ideals.