{
  "id": "1403.0666",
  "version": "v2",
  "published": "2014-03-04T02:45:45.000Z",
  "updated": "2015-06-23T20:58:58.000Z",
  "title": "Factoring the characteristic polynomial of a lattice",
  "authors": [
    "Joshua Hallam",
    "Bruce E. Sagan"
  ],
  "comment": "25 pages, 5 figures. To appear in JCTA",
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers. This method is based on the notion of a quotient of a poset which will be developed to explain this factorization. Our main theorem will give two simple conditions under which the characteristic polynomial factors with nonnegative integer roots. We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization. To our knowledge, all other theorems in this area only give conditions which imply factorization. This theorem will be used to connect the generating function for increasing spanning forests of a graph to its chromatic polynomial. We finish by mentioning some other applications of quotients of posets as well as some open questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-04T02:45:45.000Z",
      "title": "Factorization of the Characteristic Polynomial",
      "abstract": "We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our main theorem gives two simple conditions under which the characteristic polynomial factors in this way. We will see that Stanley's Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate the factorization of a polynomial related to increasing forests.",
      "comment": "23 pages, 5 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-23T20:58:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05A15"
    ],
    "keywords": [
      "factorization",
      "stanleys supersolvability theorem",
      "characteristic polynomial factors",
      "simple conditions",
      "finite lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.0666H"
    }
  }
}