{
  "id": "2103.07271",
  "version": "v1",
  "published": "2021-03-11T02:43:04.000Z",
  "updated": "2021-03-11T02:43:04.000Z",
  "title": "ZZ Polynomials of Regular $m$-tier Benzenoid Strips as Extended Strict Order Polynomials of Associated Posets -- Part 1. Proof of Equivalence",
  "authors": [
    "Johanna Langner",
    "Henryk A. Witek"
  ],
  "comment": "36 pages, submitted to MATCH Commun. Math. Comput. Chem",
  "categories": [
    "math.CO"
  ],
  "abstract": "In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial $\\text{ZZ}(\\boldsymbol{S},x)$ of a Kekul\\'ean regular $m$-tier strip $\\boldsymbol{S}$ of length $n$ and the extended strict order polynomial $\\text{E}_{\\mathcal{S}}^{\\circ}(n,x+1)$ of a certain partially ordered set (poset) $\\mathcal{S}$ associated with $\\boldsymbol{S}$. The discovered equivalence is a consequence of the one-to-one correspondence between the set $\\left\\{ K\\right\\}$ of Kekul\\'e structures of $\\boldsymbol{S}$ and the set $\\left\\{ \\mu:\\mathcal{S}\\supset\\mathcal{A}\\rightarrow\\left[\\,n\\,\\right]\\right\\}$ of strictly order-preserving maps from the induced subposets of $\\mathcal{S}$ to the interval $\\left[\\thinspace n\\thinspace\\right]$. As a result, the problems of determining the Zhang-Zhang polynomial of $\\boldsymbol{S}$ and of generating the complete set of Clar covers of $\\boldsymbol{S}$ reduce to the problem of constructing the set $\\mathcal{L}(\\mathcal{S})$ of linear extensions of the corresponding poset $\\mathcal{S}$ and studying their basic properties. In particular, the Zhang-Zhang polynomial of $\\boldsymbol{S}$ can be written in a compact form as $\\text{ZZ}(\\boldsymbol{S},x)=\\sum_{k=0}^{\\left|\\mathcal{S}\\right|}\\sum_{w\\in\\mathcal{L}(\\mathcal{S})}\\binom{\\left|\\mathcal{S}\\right|-\\text{fix}_{\\mathcal{S}}(w)}{\\,\\,k\\,\\,\\hspace{1pt}-\\text{fix}_{\\mathcal{S}}(w)}\\binom{n+\\text{des}(w)}{k}\\left(1+x\\right)^{k}$, where $\\text{des}(w)$ and $\\text{fix}_{\\mathcal{S}}(w)$ denote the number of descents and the number of fixed labels, respectively, in the linear extension $w\\in\\mathcal{L}(\\mathcal{S})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-11T02:43:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07"
    ],
    "keywords": [
      "extended strict order polynomial",
      "tier benzenoid strips",
      "zz polynomials",
      "associated posets",
      "equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}