{
  "abstract": "Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the \\emph{partial-twuality polynomials} of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality $δ$ and partial Petrie duality $τ$, on pairs $(M,A)$, where $M$ is a square matrix whose rows and columns are indexed by a finite set $V$ and $A\\subseteq V$, such that $δ^2=τ^2=(δτ)^3=id$ and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying \\(\\bullet\\) to \\((M, A)\\). In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field $\\mathbb{GF}(2)$. We prove that there exist two local operations \\(δ\\) and \\(τ\\) on \\(\\bigl((M,I_{|V|}),A\\bigr)\\) satisfying $δ^2=τ^2=(δτ)^3=id$ and $P_{\\langle \\bullet \\rangle}((M,I_{|V|}),z)=P_{\\langle \\bullet \\rangle}(M,z)$ for $\\bullet \\in \\{δ, τ, δτ, τδ, δτδ\\}$, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial \\(\\langleδτδ\\rangle\\)-polynomial with respect to an edge. This recurrence enables the computation of the partial \\(\\langleδτδ\\rangle\\)-polynomial for certain bouquets, simple graphs, and simple signed graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22273",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiaoxiang Yu"
    },
    {
      "name": "Rong-Xia Hao"
    },
    {
      "name": "Jianbing Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3zdpcya5d99mr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22273",
  "fulltext_url": null,
  "id": "3zdpcya5d99mr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22273",
  "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": "W7214055791",
  "pdf_url": "https://arxiv.org/pdf/2609.22273",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3zdpcya5d99mr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22273",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22273"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Partial-Twuality Polynomials of Paired Matrices",
  "title_text": "Partial-Twuality Polynomials of Paired Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/3zdpcya5d99mr",
  "venue": "arXiv",
  "versions": []
}