{
  "id": "math/0311230",
  "version": "v1",
  "published": "2003-11-14T01:54:09.000Z",
  "updated": "2003-11-14T01:54:09.000Z",
  "title": "M-partitions: Optimal partitions of weight for one scale pan",
  "authors": [
    "Edwin O'Shea"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the positive integers, this recurrence relation will have a generating function. The generating function will be, in some sense, the same as the generating function for counting the number of distinct binary partitions for a given integer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-11-14T01:54:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17"
    ],
    "keywords": [
      "scale pan",
      "m-partition",
      "optimal partitions",
      "positive integer",
      "generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....11230O"
    }
  }
}