{
  "id": "1409.7038",
  "version": "v1",
  "published": "2014-09-24T18:29:27.000Z",
  "updated": "2014-09-24T18:29:27.000Z",
  "title": "The number of simultaneous core partitions",
  "authors": [
    "Huan Xiong"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is $\\sum_{0\\leq k\\leq [\\frac{t}{2}]}\\frac{1}{k+1}\\binom{t}{2k}\\binom{2k}{k}$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t, t + 1, ..., t + p)$-core partitions. In particular, this verifies that Amdeberhan's conjecture is true. We also prove that the number of $(t_1,t_2,..., t_m)$-core partitions is finite if and only if gcd$(t_1,t_2,..., t_m)=1,$ which extends Anderson's result on the finiteness of the number of $(t_1,t_2)$-core partitions for coprime positive integers $t_1$ and $t_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-24T18:29:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P81"
    ],
    "keywords": [
      "simultaneous core partitions",
      "extends andersons result",
      "coprime positive integers",
      "amdeberhans conjecture",
      "generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.7038X"
    }
  }
}