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Constructive recurrences for determinants and permanents of banded Toeplitz matrices — arXiv:2609.13674

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.13674
License (as archived): CC0-1.0
Archived: 2026-10-16T01:25:41.374Z
SHA-256 of the archived text: 63efa83fdfe1682019bbe426553e97321aa3f1317b974eaf12a8cc4c927cd3d5
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Constructive recurrences for determinants and permanents of banded Toeplitz matrices — arXiv:2609.13674 — arXiv (en), https://arxiv.org/abs/2609.13674 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bfbca5b7d1df (Metraipolis archive dc_bfbca5b7d1df, sha256 63efa83fdfe1682019bbe426553e97321aa3f1317b974eaf12a8cc4c927cd3d5)

BibTeX

@misc{dcbfbca5b7d1df,
  title = {Constructive recurrences for determinants and permanents of banded Toeplitz matrices — arXiv:2609.13674},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.13674},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bfbca5b7d1df; sha256 63efa83fdfe1682019bbe426553e97321aa3f1317b974eaf12a8cc4c927cd3d5},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.13674. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-15 in math.CO).
Categorie arXiv: Rings and Algebras (math.RA) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.13674 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T00:39:30.272Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).