# Rook placements and Jordan forms of upper-triangular nilpotent matrices

## TL;DR

This paper provides a combinatorial formula for counting upper-triangular nilpotent matrices over finite fields by their Jordan forms, linking rook placements and Young tableaux to refine enumeration methods.

## Contribution

It introduces a new combinatorial formula for F_____(q) using standard Young tableaux and explores rook placements to refine this enumeration.

## Key findings

- Derived a weighted sum formula for F_____(q)
- Established a connection between matrices and non-attacking rook placements
- Provided a combinatorial refinement of the counting formula

## Abstract

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).