The set of n by n upper-triangular nilpotent matrices with entries in a finite field 𝔽q has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number Fλ(q) of matrices of Jordan type λ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for Fλ(q).

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Published in The Electronic Journal of Combinatorics, v. 25, issue 1, paper #P1.68, p. 1-25.

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Simons Collaboration Grant 429920.

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