Partitions of matrix spaces with an application to q-rook polynomials

Heide Gluesing-Luerssen, Alberto Ravagnani

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


We study the row-space partition and the pivot partition on the matrix space Fq n×m. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear rank-metric codes. We then generalize the Singleton-like bound for rank-metric codes, and introduce two new concepts of code extremality. Both of them generalize the notion of MRD code and are preserved by trace-duality. Moreover, codes that are extremal according to either notion satisfy strong rigidity properties analogous to those of MRD codes. As an application of our results to combinatorics, we give closed formulas for the q-rook polynomials associated with Ferrers diagram boards. Moreover, we exploit connections between matrices over finite fields and rook placements to prove that the number of matrices of rank r over Fq supported on a Ferrers diagram is a polynomial in q, whose degree is strictly increasing in r. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions.

Original languageEnglish
Article number103120
JournalEuropean Journal of Combinatorics
StatePublished - Oct 2020

Bibliographical note

Funding Information:
The author was partially supported by grant # 422479 from the Simons Foundation, USA.The author was supported by the Swiss National Science Foundation through grant # P2NEP2_168527 and by the Marie Curie, United Kingdom Research Grants Scheme, grant # 740880.

Publisher Copyright:
© 2020 Elsevier Ltd

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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