Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices

Abstract

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 $F_\lambda(q)$.