{
  "id": "2409.12934",
  "version": "v1",
  "published": "2024-09-19T17:45:21.000Z",
  "updated": "2024-09-19T17:45:21.000Z",
  "title": "On $e$-positivity of trees and connected partitions",
  "authors": [
    "Foster Tom"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that a tree with a vertex of degree at least five must be missing a connected partition of some type and therefore its chromatic symmetric function cannot be $e$-positive. We prove that this also holds for a tree with a vertex of degree four as long as it is not adjacent to any leaf. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-19T17:45:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05C05",
      "05C15"
    ],
    "keywords": [
      "connected partition",
      "positivity",
      "positive chromatic symmetric function",
      "van willigenburg"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}