{
  "id": "2305.15554",
  "version": "v1",
  "published": "2023-05-24T20:41:33.000Z",
  "updated": "2023-05-24T20:41:33.000Z",
  "title": "Unit-Interval Parking Functions and the Permutohedron",
  "authors": [
    "Lucas Chaves Meyles",
    "Pamela E. Harris",
    "Richter Jordaan",
    "Gordon Rojas Kirby",
    "Sam Sehayek",
    "Ethan Spingarn"
  ],
  "comment": "15 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Unit-interval parking functions are subset of parking functions in which cars park at most one spot away from their preferred parking spot. In this paper, we characterize unit-interval parking functions by understanding how they decompose into prime parking functions and count unit-interval parking functions when exactly $k<n$ cars do not park in their preference. This count yields an alternate proof of a result of Hadaway and Harris establishing that unit-interval parking functions are enumerated by the Fubini numbers. Then, our main result, establishes that for all integers $0\\leq k<n$, the unit-interval parking functions of length $n$ with displacement $k$ are in bijection with the $k$-dimensional faces of the permutohedron of order $n$. We conclude with some consequences of this result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-24T20:41:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "permutohedron",
      "count unit-interval parking functions",
      "spot away",
      "characterize unit-interval parking functions",
      "cars park"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}