{
  "abstract": "For fixed nonnegative integers $m_1,m_2$, let $A_n=(a_{j-i})_{i,j=1}^n$ be the leading $n\\times n$ section of a Toeplitz matrix with lower and upper semibandwidths $m_1$ and $m_2$. We give two constructive Laplace-expansion methods for scalar recurrences of $\\det A_n$ and $\\perm A_n$. The increasing-rows method eliminates a fixed family of boundary cofactors and gives recurrence order at most $d=\\binom{m_1+m_2}{m_1}$ for both sequences. The row-column method closes normalized boundary minors recursively and packages them in a sparse transfer matrix. Its reachable states are classified exactly: level $j$ is indexed by a pair of $j$-subsets of $[m_1]$ and $[m_2]$. Hence the transfer dimension is $d$, and we obtain an explicit formula for the number of nonzero transitions. For determinants, the complementary cofactors of the increasing-rows construction are coordinates of the classical compound companion representation. The independently constructed row-column transfer has the Widom characteristic polynomial and is generically similar to the compound transfer. Thus the order $d$ recurrence is generically minimal for the unrestricted fixed-band determinant family. For permanents the same state graph gives the binomial upper bound, without a general minimality claim. The pentadiagonal case recovers Sweet's order-six determinant recurrence and its permanent analogue, while the one-superdiagonal family gives closed scalar recurrences and rational generating functions. Position-dependent band weights preserve the finite state graph but replace the constant transfer by a cocycle. For cyclic determinants, Fourier diagonalization produces all subset products of the symbol roots and a generically minimal annihilator of degree $2^{m_1+m_2}$, corresponding to the passage from one exterior degree to the full exterior algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13674",
  "author_total": 2,
  "authors": [
    {
      "name": "Max A. Alekseyev"
    },
    {
      "name": "Dmitry I. Khomovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fyj100ryk4cs0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13674",
  "fulltext_url": null,
  "id": "fyj100ryk4cs0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13674",
  "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",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7213252130",
  "pdf_url": "https://arxiv.org/pdf/2609.13674",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fyj100ryk4cs0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13674",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13674"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Constructive recurrences for determinants and permanents of banded Toeplitz matrices",
  "title_text": "Constructive recurrences for determinants and permanents of banded Toeplitz matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/fyj100ryk4cs0",
  "venue": "arXiv",
  "versions": []
}