{
  "id": "2102.00412",
  "version": "v1",
  "published": "2021-01-31T08:37:47.000Z",
  "updated": "2021-01-31T08:37:47.000Z",
  "title": "One Curious Identity Counting Graceful Labelings",
  "authors": [
    "Nikolai Beluhov"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $a$ and $b$ be positive integers with prime factorisations $a = p_1^np_2^n$ and $b = q_1^nq_2^n$. We prove that the number of essentially distinct $\\alpha$-graceful labelings of the complete bipartite graph $K_{a, b}$ equals the alternating sum of fourth powers of binomial coefficients $(-1)^n[\\binom{2n}{0}^4 - \\binom{2n}{1}^4 + \\binom{2n}{2}^4 - \\binom{2n}{3}^4 + \\cdots + \\binom{2n}{2n}^4]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-31T08:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05C30",
      "05C78"
    ],
    "keywords": [
      "curious identity counting graceful labelings",
      "complete bipartite graph",
      "prime factorisations",
      "fourth powers",
      "binomial coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}