Geodesic
Matrices with restricted entries and q-analogues of permutations (extended abstract)
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011).

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We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions. 
@article{DMTCS_2011_special_260_a54,
     author = {Lewis, Joel Brewster and Liu, Ricky Ini and Morales, Alejandro H. and Panova, Greta and Sam, Steven V and Zhang, Yan},
     title = {Matrices with restricted entries and q-analogues of permutations (extended abstract)},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)},
     year = {2011},
     doi = {10.46298/dmtcs.2941},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2941/}
}
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%A Liu, Ricky Ini
%A Morales, Alejandro H.
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%A Sam, Steven V
%A Zhang, Yan
%T Matrices with restricted entries and q-analogues of permutations (extended abstract)
%J Discrete mathematics & theoretical computer science
%D 2011
%V DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
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Lewis, Joel Brewster; Liu, Ricky Ini; Morales, Alejandro H.; Panova, Greta; Sam, Steven V; Zhang, Yan. Matrices with restricted entries and q-analogues of permutations (extended abstract). Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011). doi : 10.46298/dmtcs.2941. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2941/

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