{
  "id": "2101.02131",
  "version": "v1",
  "published": "2021-01-06T16:53:01.000Z",
  "updated": "2021-01-06T16:53:01.000Z",
  "title": "Theorems and Conjectures on Some Rational Generating Functions",
  "authors": [
    "Richard P. Stanley"
  ],
  "comment": "24 pages, two figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $I_n(x)=\\prod_{i=1}^n \\left( 1+x^{F_{i+1}}\\right)$, where $F_{i+1}$ denotes a Fibonacci number. Let $v_r(n)$ denote the sum of the $r$th powers of the coefficients of $I_n(x)$. Our prototypical result is that $\\sum_{n\\geq 0} v_2(n)x^n= (1-2x^2)/(1-2x-2x^2+2x^3)$. We give many related results and conjectures. A certain infinite poset $\\mathfrak{F}$ is naturally associated with $I_n(x)$. We discuss some combinatorial properties of $\\mathfrak{F}$ and a natural generalization, including a symmetric function that encodes the flag $h$-vector of $\\mathfrak{F}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-06T16:53:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05E05",
      "05A10",
      "06A07"
    ],
    "keywords": [
      "rational generating functions",
      "conjectures",
      "fibonacci number",
      "symmetric function",
      "natural generalization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}