{
  "abstract": "Richard Stanley asked whether every tree $T$ is determined up to isomorphism by its chromatic symmetric function $X_T$. We approach this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1, p_2, \\dots$ and studying the invariant $Φ_T = \\left.\\frac{\\partial {X_T}}{\\partial p_1}\\right\\rvert_{p_1 = 0}$. We prove that $Φ_T$ distinguishes, among proper trees, every proper tree in which any non-leaf vertex with two or more non-leaf neighbors has a number of leaf neighbors occurring at no other non-leaf vertex. We give an equivalent formulation of Stanley's question in terms of $Φ_T$. Finally, we show that counts of spanning forests with at most $t$ edges cannot distinguish all $k$-vertex trees when $t<\\left\\lfloor k/2\\right\\rfloor$, for every $k\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13464",
  "author_total": 1,
  "authors": [
    {
      "name": "Saad A. Awan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59dyr7q9k2sq7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13464",
  "fulltext_url": null,
  "id": "59dyr7q9k2sq7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13464",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213263805",
  "pdf_url": "https://arxiv.org/pdf/2609.13464",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59dyr7q9k2sq7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.13464",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13464"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "First-Derivative Chromatic Symmetric Reconstruction For Proper Trees",
  "title_text": "First-Derivative Chromatic Symmetric Reconstruction For Proper Trees",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/59dyr7q9k2sq7",
  "venue": "arXiv",
  "versions": []
}