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We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free d-generator Lie rings \(\mathfrak {f}_{c,d}\) of nilpotency class c for all \(c\leqslant 2\) and for \((c,d)\in \{(3,3),(3,2),(4,2)\}\). We apply our computations to obtain information about \(\mathfrak {p}\)-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings