Abstract

Given an ordered triple of positive integers (n,r,b), where 1≤b≤nr, does there exist a matrix of size r×n with exactly b invertible submatrices of size r×r? Such a matrix is called an (n,r,b)-matrix. This question is a stronger version of an open problem in matroid theory raised by Dominic Welsh. In this paper, we prove that an (n,r,b)-matrix exists when the corank satisfies n−r≤3, unless (n,r,b)=(6,3,11). Furthermore, we show that an (n,r,b)-matrix exists when the rank r is large relative to the corank n−r.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.