Classification of small $(0,1)$ matrices
Abstract
Denote by $A_n$ the set of square $(0,1)$ matrices of order $n$. The set $A_n$, $n\le8$, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular $(0,1)$ matrices of order 8 is 10160459763342013440. Let $D_n$, $S_n$ denote the set of absolute determinant values and Smith normal forms of matrices from $A_n$. Denote by $a_n$ the smallest integer not in $D_n$. The sets $\mathcal{D}_9$ and $\mathcal{S}_9$ are obtained; especially, $a_9=103$. The lower bounds for $a_n$, $10\le n\le 19$, (exceeding the known lower bound $a_n\ge 2f_{n-1}$, where $f_n$ is $n$th Fibonacci number) are obtained. Row/permutation equivalence classes of $A_n$ correspond to bipartite graphs with $n$ black and $n$ white vertices, and so the other applications of the classification are possible.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- November 2005
- DOI:
- 10.48550/arXiv.math/0511636
- arXiv:
- arXiv:math/0511636
- Bibcode:
- 2005math.....11636Z
- Keywords:
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- Mathematics - Combinatorics;
- 15A21;
- 15A36;
- 11Y55
- E-Print:
- 45 pages. submitted to LAA