{
  "id": "0805.1467",
  "version": "v5",
  "published": "2008-05-12T10:17:34.000Z",
  "updated": "2015-07-19T11:17:17.000Z",
  "title": "Marked partitions and combinatorial analogs of the Sylvester identity",
  "authors": [
    "F. V. Weinstein"
  ],
  "comment": "10 pages The text was completely revised. Compared to the previous version it contains new proofs and results",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n>0$ be an integer. A partition of degree $n$ is a decomposition $n=i_1+i_2+\\dots+i_q$, where $0<i_1\\leq i_2\\leq \\dots\\leq i_q$ are integers. The numbers $i_1,i_2,\\dots,i_q$ are called the parts of the partition. Let $k\\>1$ and $\\lambda\\>0$ be integers. The partition is said to be a $(\\lambda,k)$--partition if $i_1\\>k$ and $i_{a+1}-i_a\\>\\lambda$ for all $a\\in[1,q-1]$. The main result of the article is a construction of the bijective maps for $\\lambda=2$ or $\\lambda=3$ and any $k$ between the set of $(1,k)$-partitions of a given degree and the sets of the $(\\lambda,k)$--partitions of the same degree, some special parts of which are marked. These bijections lead to combinatorial identities. I construct one such map for $\\lambda=3$, and another one for simultaneously $\\lambda=2$ and $\\lambda=3$. For $\\lambda=3$, the maps thus defined are different. The corresponding identity for $\\lambda=3$ can be presented analytically. In this form, for $k=1$, it coincides with the classical Sylvester identity. For $\\lambda=2$, the result also implies an interesting identity.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-06-30T15:23:31.000Z",
      "title": "One identity for integer partitions and its bijective proofs",
      "abstract": "The main result of the note is a combinatorial identity that expresses the partition's quantity of natural $n$ with $q$ distinct parts by means of the partitions of $n$, for which the differences between parts are not less than either $\\lb=2$, or $\\lb=3$. Such partitions are called $\\lb$-partitions. For them is introduced a notion of index - a non negative integer that depends on $\\lb$. One corollary of the identity is the formula $d(n)=\\sum_{\\alpha=0}^\\infty p_\\lb(n,\\alpha)2^\\alpha$, where $d(n)$ is the partition's quantity of $n$ with distinct parts and $p_\\lb(n,\\alpha)$ is the $\\lb$-partition's quantity of $n$, index of which equals to $\\alpha$. For $\\lb=3$ the identity turns to be equivalent to the famous Sylvester formula and gives a new combinatorial interpretation for it. Two bijective proofs of the main result are provided: one for $\\lb=3$ and another one for $\\lb=2$ and $\\lb=3$ simultaneously.",
      "comment": "7 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-07-19T11:17:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "05A19"
    ],
    "keywords": [
      "bijective proofs",
      "integer partitions",
      "partitions quantity",
      "main result",
      "distinct parts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0805.1467W"
    }
  }
}