Counting Lie ideals of niltriangular matrices

We give a formula for the number of ideals of the Lie algebra of strictly lower triangular $n\times n$ matrices over ${\mathbb{F}}_q$. A contraction bijection transforms Gagnon’s sum into a weighted enumeration of nonnesting partitions, with antichains of intervals chosen independently in each block. The block weights are the inversion polynomials for $321$-avoiding permutations. Combining the known Stieltjes continued fraction for these polynomials with the enumeration of nonnesting partitions by block sizes yields a formula involving $n-1$ coefficient extractions, valid for every prime power $q$.