{
  "id": "2112.03788",
  "version": "v2",
  "published": "2021-12-07T16:04:06.000Z",
  "updated": "2024-05-30T13:55:04.000Z",
  "title": "Enumerating Matroids and Linear Spaces",
  "authors": [
    "Matthew Kwan",
    "Ashwin Sah",
    "Mehtaab Sawhney"
  ],
  "doi": "10.5802/crmath.423",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\\sqrt 3/2-3}(1+\\sqrt 3)/2$. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: the numbers of rank-1 and rank-2 matroids on a ground set of size $n$ have exact representations in terms of well-known combinatorial functions, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant $r\\ge 4$ there are $(e^{1-r}n+o(n))^{n^{r-1}/r!}$ rank-$r$ matroids on a ground set of size $n$. In our proof, we introduce a new approach for bounding the number of clique decompositions of a complete graph, using quasirandomness instead of the so-called entropy method that is common in this area.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-05-30T13:55:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear spaces",
      "enumerating matroids",
      "ground set",
      "well-known combinatorial functions",
      "van der hofstad"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}