{
  "id": "2108.07042",
  "version": "v1",
  "published": "2021-08-16T12:23:12.000Z",
  "updated": "2021-08-16T12:23:12.000Z",
  "title": "On the minimum size of subset and subsequence sums in integers",
  "authors": [
    "Jagannath Bhanja",
    "Ram Krishna Pandey"
  ],
  "comment": "18 pages, comments are welcome!",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{A}$ be a finite sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\\mathcal{A}$. The sum of all terms of a subsequence of $\\mathcal{A}$ is called a subsequence sum of $\\mathcal{A}$. For a fixed nonnegative integer $\\alpha \\leq rk$, let $\\Sigma_{\\alpha} (r,\\mathcal{A})$ be the set of all subsequence sums of $\\mathcal{A}$ that correspond to the subsequences of length $\\alpha$ or more. When $r=1$, we call the set of subsequence sums as the set of subset sums and we denote it by $\\Sigma_{\\alpha} (A)$ instead of $\\Sigma_{\\alpha} (1,\\mathcal{A})$. In this article, we give optimal lower bounds for the sizes of $\\Sigma_{\\alpha} (A)$ and $\\Sigma_{\\alpha} (r,\\mathcal{A})$. As special cases, we also obtain some already known results in this study.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-16T12:23:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P70",
      "11B75",
      "11B13"
    ],
    "keywords": [
      "subsequence sum",
      "minimum size",
      "optimal lower bounds",
      "distinct integers",
      "finite sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}