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Description

The set of upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by the partitions $\lambda\vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ as a weighted sum over standard Young tableaux. We then discuss connections between these matrices, non-attacking rook placements, and set partitions, which lead to a refinement of the formula for $F_\lambda(q)$.