Rook placements and Jordan forms of upper-triangular nilpotent matrices (Q1753039)

Summary: The set of \(n\) by \(n\) upper-triangular nilpotent matrices with entries in a finite field \(\mathbb{F}_q\) has Jordan canonical forms indexed by 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 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 \(F_\lambda(q)\).