{
  "id": "1804.09091",
  "version": "v1",
  "published": "2018-04-24T15:21:54.000Z",
  "updated": "2018-04-24T15:21:54.000Z",
  "title": "On the Polynomiality of moments of sizes for random $(n, dn\\pm 1)$-core partitions with distinct parts",
  "authors": [
    "Huan Xiong",
    "Wenston J. T. Zang"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Amdeberhan's conjectures on $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-core partitions with distinct parts, respectively. Let $X_{s,t}$ be the size of a uniform random $(s,t)$-core partition with distinct parts when $s$ and $t$ are coprime to each other. Some explicit formulas for moments of $X_{n,n+1}$ and $X_{2n+1,2n+3}$ were given by Zaleski and Zeilberger. Zaleski also studied the expectations and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we prove several polynomiality results for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ in this paper. In particular, we show that these $k$-th moments are asymptotically some polynomials of n with degrees at most $2k$, when $d$ is given and $n$ tends to infinity. The explicit formulas for the expectations $\\mathbb{E} [X_{n,dn+1}]$ and $\\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves Zaleski's polynomiality conjecture on the $k$-th moment of $X_{n,dn-1}$. Moreover, when $d=1$, we show that the $k$-th moment $\\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\\left(n^2/10\\right)^k$ when $n$ tends to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-24T15:21:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P81"
    ],
    "keywords": [
      "distinct parts",
      "core partition",
      "th moment",
      "explicit formulas",
      "zaleskis polynomiality conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}