Building matrices with prescribed size and number of invertible submatrices
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2014
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| _version_ | 1866916803664412672 |
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| author | Fan, Edward S. T. Wong, Tony W. H. |
| author_facet | Fan, Edward S. T. Wong, Tony W. H. |
| contents | Given an ordered triple of positive integers $(n,r,b)$, where $1\leq b\leq\binom{n}{r}$, does there exist a matrix of size $r\times n$ with exactly $b$ invertible submatrices of size $r\times r$? 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 $n-r\leq3$, 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 $n-r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1402_6048 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Building matrices with prescribed size and number of invertible submatrices Fan, Edward S. T. Wong, Tony W. H. Combinatorics 15A03, 05B35, 05A05 Given an ordered triple of positive integers $(n,r,b)$, where $1\leq b\leq\binom{n}{r}$, does there exist a matrix of size $r\times n$ with exactly $b$ invertible submatrices of size $r\times r$? 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 $n-r\leq3$, 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 $n-r$. |
| title | Building matrices with prescribed size and number of invertible submatrices |
| topic | Combinatorics 15A03, 05B35, 05A05 |
| url | https://arxiv.org/abs/1402.6048 |