Rook placements and Jordan forms of upper-triangular nilpotent matrices

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1 Scopus citations


The set of n by n upper-triangular nilpotent matrices with entries in a finite field Fq has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number Fλ (q) of matrices of Jordan type λ 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 Fλ (q).

Original languageEnglish
Article number#P1.68
JournalElectronic Journal of Combinatorics
Issue number1
StatePublished - Mar 29 2018

Bibliographical note

Publisher Copyright:
© 2018, Australian National University. All rights reserved.


  • Finite fields
  • Jordan form
  • Nilpotent matrices
  • Rook placements
  • Set partitions
  • Young tableaux

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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