{
  "id": "2401.06095",
  "version": "v1",
  "published": "2024-01-11T18:19:36.000Z",
  "updated": "2024-01-11T18:19:36.000Z",
  "title": "On the Structure and Generators of the $n$th-order Chromatic Algebra",
  "authors": [
    "Ethan Yi-Heng Liu"
  ],
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "This work investigates the intrinsic properties of the chromatic algebra, introduced by Fendley and Krushkal as a framework to study the chromatic polynomial. We prove that the dimension of the $n$th-order chromatic algebra is the $2n$th Riordan number, which exhibits exponential growth. We find a generating set of size $\\binom{n}{2}$, and we provide a procedure to construct the basis from the generating set. We additionally provide proofs for fundamental facts about this algebra that appear to be missing from the literature. These include determining a representation of the chromatic algebra as noncrossing planar partitions and expanding the chromatic relations to include an edge case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-11T18:19:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E16"
    ],
    "keywords": [
      "th-order chromatic algebra",
      "generators",
      "th riordan number",
      "generating set",
      "chromatic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}