  The set of n by n upper-triangular nilpotent matrices with entries in a
finite field F_q has Jordan canonical forms indexed by partitions lambda of 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 also study a connection between these matrices and non-attacking rook
placements, which leads to a refinement of the formula for F_\lambda(q).
