{
  "id": "2501.00335",
  "version": "v1",
  "published": "2024-12-31T08:14:53.000Z",
  "updated": "2024-12-31T08:14:53.000Z",
  "title": "Bijections around Springer numbers",
  "authors": [
    "Shaoshi Chen",
    "Yang Li",
    "Zhicong Lin",
    "Sherry H. F. Yan"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type $B$ analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a ``hard'' bijection with snakes. Callan conjectured in 2012 and Han--Kitaev--Zhang proved recently that rc-invariant alternating permutations are counted by Springer numbers. Very recently, Chen--Fang--Kitaev--Zhang investigated multi-dimensional permutations and proved that weakly increasing $3$-dimensional permutations are also counted by Springer numbers. In this work, we construct a sequence of ``natural'' bijections linking the above four combinatorial objects.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-31T08:14:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "springer numbers enumerate",
      "combinatorial objects",
      "rc-invariant alternating permutations",
      "multi-dimensional permutations",
      "labeled ballot paths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}