Matrices with restricted entries and q-analogues of permutations

Details Matrices with restricted entries and q-analogues of permutations Matrices with restricted entries and q-analogues of permutations

2010-11-20

arxiv.org/abs/1011.4539v3

J. Comb. 2 (2011), no. 3, 355-396

Mathematics

Combinatorics

29 pages, 2 figures, v2: one additional result, some formulas simplified, and a new reference; v3: corrected typos

Scientific paper

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, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

Affiliated with

Also associated with

No associations

Rating

Matrices with restricted entries and q-analogues of permutations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Matrices with restricted entries and q-analogues of permutations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Matrices with restricted entries and q-analogues of permutations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-21551