{
  "abstract": "This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent--determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from $\\binom{2m}{m}$ to the Catalan number $C_{m+1}$; primitive symplectic weight compression then leaves $(3^m+1)/2$ distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension $\\binom{2m}{m}/2$, the one-step compound transfer exchanges the halves, and its two-step restriction has $3^{m-1}$ generic ternary modes. Thus the full skew bound is $2\\cdot3^{m-1}$. Widom--Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a Pólya--Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion---and separately Toeplitz diagonal-wise conversion---occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound $(\\binom{2m}{m}+2^m)/2$, while cyclic defect-sector pairing gives $(4^m+\\binom{2m}{m})/2$. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23224",
  "author_total": 2,
  "authors": [
    {
      "name": "Max A. Alekseyev"
    },
    {
      "name": "Dmitry I. Khomovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgwvat904vdht.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23224",
  "fulltext_url": null,
  "id": "dgwvat904vdht",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23224",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214019416",
  "pdf_url": "https://arxiv.org/pdf/2609.23224",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgwvat904vdht.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23224",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23224"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents",
  "title_text": "Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/dgwvat904vdht",
  "venue": "arXiv",
  "versions": []
}