Geodesic
q-Rook placements and Jordan forms of upper-triangular nilpotent matrices
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

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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 $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$ F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements. 
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     author = {Yip, Martha},
     title = {q-Rook placements and {Jordan} forms of upper-triangular nilpotent matrices},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)},
     year = {2013},
     doi = {10.46298/dmtcs.2362},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2362/}
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Yip, Martha. q-Rook placements and Jordan forms of upper-triangular nilpotent matrices. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013). doi : 10.46298/dmtcs.2362. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2362/

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