https://hal.inria.fr/hal-01229691Yip, MarthaMarthaYipUniversity of Pennsylvania [Philadelphia]q-Rook placements and Jordan forms of upper-triangular nilpotent matricesHAL CCSD2013q-rook placementsJordan canonical formnilpotent matricesset partitions[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Monteil, AlainAlain Goupil and Gilles Schaeffer2015-11-17 10:20:012017-03-07 15:23:382015-11-17 10:24:59enConference papershttps://hal.inria.fr/hal-01229691/document10.46298/dmtcs.2362application/pdf1The 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.