{
  "abstract": "Our combinatorial analysis is motivated by the PDE dynamics \\begin{equation} \\mathbf{u_t} = \\mathbf{u_{xx}} + \\mathbf{g}(\\mathbf{u}),\\qquad 0<\\mathbf{x}<1, \\end{equation} of solutions $\\mathbf{u}=\\mathbf{u}(\\mathbf{t},\\mathbf{x}),\\ \\mathbf{t}\\geq 0$, under Neumann boundary conditions. For dissipative nondegenerate nonlinearities $\\mathbf{g}$, the global attractors $\\mathcal{A}=\\mathcal{A}_\\mathbf{g}$ of the PDE can then be classified by the orderings of their $2n+1$ equilibria $\\mathbf{v}$ at the boundaries $\\mathbf{x}=0,1$. We encode the boundary orders as Hamiltonian Sturm permutations. The name ''Sturm'' refers to nodal properties of PDE solutions $\\mathbf{u}(\\mathbf{t},\\mathbf{x})$. ''Hamiltonian'' refers to the second order pendulum ODE for equilibria $\\mathbf{v}(\\mathbf{x})$: \\begin{equation} 0 = \\mathbf{v_{xx}} + \\mathbf{g}(\\mathbf{v}). \\end{equation} We determine the generating function $a(z)=\\sum_n a_nz^n$ for the counts $a_n$ of Hamiltonian Sturm permutations. For $n\\rightarrow\\infty$, this provides explicit asymptotics of $a_n$. We refine these counts as $a_n=\\sum b_{rq}$. Here $b_{rq}$ counts Hamiltonian Sturm permutations with $2r+1$ spatially homogeneous equilibria and $2q$ spatially non-homogeneous equilibria, such that $r+q=n$. We also determine the explicit generating function $b(x,y)=\\sum_{r,q} b_{rq}x^ry^q$. This implies asymptotically Gaussian distributions of the probabilities $p_{nr}=b_{rq}/a_n$ with $r+q=n$, asymptotically for large $n$. We derive asymptotics for means and variances, with error estimates of order $1/n$. All asymptotics are based on work by Flajolet and Sedgewick. We conclude with numerical illustrations and remarks on nonlinearities $\\mathbf{g}(\\mathbf{u},\\mathbf{u_x})$ under periodic boundary conditions $\\mathbf{x}\\in\\mathbb{S}^1=\\mathbb{R}/2\\mathbb{Z}$, where rotating waves arise.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03611",
  "author_total": 2,
  "authors": [
    {
      "name": "Bernold Fiedler"
    },
    {
      "name": "Carlos Rocha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dndmk3cf1fhw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03611",
  "fulltext_url": null,
  "id": "2dndmk3cf1fhw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03611",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220386697",
  "pdf_url": "https://arxiv.org/pdf/2610.03611",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dndmk3cf1fhw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03611",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03611"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions",
  "title_text": "Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/2dndmk3cf1fhw",
  "venue": "arXiv",
  "versions": []
}