{
  "abstract": "We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at $t=1$ and show that the power-sum-indexed operators $Θ_{\\mathsf p_k}|_{t=1}$ agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing $\\widetilde{h}_a$ for the modified Macdonald function indexed by the single row $(a)$ we establish that $Θ_{\\mathsf p_μ}\\widetilde{h}_a|_{t=1}$ is, up to a normalization, the single-row Macdonald cumulant of Dolęga. We use these results to show that $Θ_{\\mathsf p_μ}\\widetilde{h}_a|_{t=1}$ and $Θ_{\\mathsf e_λ}\\widetilde{h}_a|_{t=1}$ are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dolęga when all shapes are single rows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29957",
  "author_total": 2,
  "authors": [
    {
      "name": "Jim Haglund"
    },
    {
      "name": "Vasu Tewari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/900386krcnzhk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29957",
  "fulltext_url": null,
  "id": "900386krcnzhk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29957",
  "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": "W7214290369",
  "pdf_url": "https://arxiv.org/pdf/2609.29957",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/900386krcnzhk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29957",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29957"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Theta operators at $t=1$, Macdonald cumulants, and LLT positivity",
  "title_text": "Theta operators at t=1, Macdonald cumulants, and LLT positivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/900386krcnzhk",
  "venue": "arXiv",
  "versions": []
}