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arXiv · 2609.09514

Counting Lie ideals of niltriangular matrices

N. D. Khodyunya
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Abstract

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 $q$.

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N. D. Khodyunya. 2026-09-08. Counting Lie ideals of niltriangular matrices. https://arxiv.org/abs/2609.09514

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