{
  "id": "0806.0434",
  "version": "v2",
  "published": "2008-06-03T05:05:01.000Z",
  "updated": "2008-06-05T02:07:30.000Z",
  "title": "Circular Peaks and Hilbert Series",
  "authors": [
    "Pierre Bouchard",
    "Jun Ma",
    "Yeong-Nan Yeh"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The circular peak set of a permutation $\\sigma$ is the set $\\{\\sigma(i)\\mid \\sigma(i-1)<\\sigma(i)>\\sigma(i+1)\\}$. Let $\\mathcal{P}_n$ be the set of all the subset $S\\subseteq [n]$ such that there exists a permutation $\\sigma$ which has the circular set $S$. We can make the set $\\mathcal{P}_n$ into a poset $\\mathscr{P}_n$ by defining $S\\preceq T$ if $S\\subseteq T$ as sets. In this paper, we prove that the poset $\\mathscr{P}_n$ is a simplicial complex on the vertex set $[3,n]$. We study the $f$-vector, the $f$-polynomial, the reduced Euler characteristic, the M$\\ddot{o}$bius function, the $h$-vector and the $h$-polynomial of $\\mathscr{P}_n$. We also derive the zeta polynomial of $\\mathscr{P}_n$ and give the formula for the number of the chains in $\\mathscr{P}_n$. By the poset $\\mathscr{P}_n$, we define two algebras $\\mathcal{A}_{\\mathscr{P}_n}$ and $\\mathcal{B}_{\\mathscr{P}_n}$. We consider the Hilbert polynomials and the Hilbert series of the algebra $\\mathcal{A}_{\\mathscr{P}_n}$ and $\\mathcal{B}_{\\mathscr{P}_n}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-05T02:07:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert series",
      "circular peak set",
      "hilbert polynomials",
      "circular set",
      "zeta polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.0434B"
    }
  }
}