{
  "id": "2210.11858",
  "version": "v1",
  "published": "2022-10-21T10:24:53.000Z",
  "updated": "2022-10-21T10:24:53.000Z",
  "title": "Tight Lower Bound for Pattern Avoidance Schur-Positivity",
  "authors": [
    "Avichai Marmor"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a set of permutations (patterns) $\\Pi$ in $S_k$, consider the set of all permutations in $S_n$ that avoid all patterns in $\\Pi$. An important problem in current algebraic combinatorics is to find pattern sets $\\Pi$ such that the corresponding quasi-symmetric function is symmetric for all $n$. Recently, Bloom and Sagan proved that for any $k \\ge 4$, the size of such $\\Pi$ must be at least $3$ unless $\\Pi \\subseteq \\{[1, 2, \\dots, k],\\; [k, \\dots, 1]\\}$, and asked for a general lower bound. We prove that the minimal size of such $\\Pi$ is exactly $k - 1$. The proof applies a new generalization of a theorem of Bose from extremal combinatorics. This generalization is proved using the multilinear polynomial approach of Alon, Babai and Suzuki to the extension by Ray-Chaudhuri and Wilson to Bose's theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-21T10:24:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tight lower bound",
      "pattern avoidance schur-positivity",
      "multilinear polynomial approach",
      "current algebraic combinatorics",
      "general lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}