{
  "id": "math/0511636",
  "version": "v1",
  "published": "2005-11-25T20:23:39.000Z",
  "updated": "2005-11-25T20:23:39.000Z",
  "title": "Classification of small $(0,1)$ matrices",
  "authors": [
    "Miodrag Živković"
  ],
  "comment": "45 pages. submitted to LAA",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-25T20:23:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A21",
      "15A36",
      "11Y55"
    ],
    "keywords": [
      "classification",
      "permutation equivalence classes enabling derivation",
      "lower bound",
      "row/column permutation equivalence classes enabling",
      "th fibonacci number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11636Z"
    }
  }
}