{
  "id": "2409.15519",
  "version": "v1",
  "published": "2024-09-23T20:13:32.000Z",
  "updated": "2024-09-23T20:13:32.000Z",
  "title": "On the $f$-vectors of flow polytopes for the complete graph",
  "authors": [
    "William T. Dugan"
  ],
  "comment": "19 pages + appendix, 5 figures, 2 tables",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Chan-Robbins-Yuen polytope ($CRY_n$) of order $n$ is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow $(1,0,0, \\ldots , 0, -1)$. The volume and lattice points of this polytope have been actively studied, however its face structure has received less attention. We give generating functions and explicit formulas for computing the $f$-vector by using Hille's (2003) result bijecting faces of a flow polytope to certain graphs, as well as Andresen-Kjeldsen's (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes of the complete graph having arbitrary (non-negative) netflow vectors and recover the $f$-vector of the Tesler polytope of M\\'esz\\'aros--Morales--Rhoades (2017).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-23T20:13:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C21",
      "52B05",
      "05A15",
      "05A19",
      "06A07",
      "52B20"
    ],
    "keywords": [
      "flow polytope",
      "directed complete graph",
      "face structure",
      "lattice points",
      "doubly stochastic matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}