Proofs of several OEIS conjectures on determinants and permanents

Sela Fried

arxiv: 2606.09913 · v1 · pith:VIEA7Q3Wnew · submitted 2026-06-06 · 🧮 math.CO

Proofs of several OEIS conjectures on determinants and permanents

Sela Fried This is my paper

Pith reviewed 2026-06-27 19:35 UTC · model grok-4.3

classification 🧮 math.CO

keywords OEIS conjecturesdeterminantspermanentsToeplitz matricescross matricesKronecker powersclosed-form formulasLU-factorizations

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The pith

Several OEIS conjectures on determinants and permanents of Toeplitz, cross, and Kronecker matrices are proved via row operations and block formulas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes proofs for multiple conjectures listed in the OEIS about the determinants and permanents of matrices with special structures. These include Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices whose entries involve powers of differences. Standard linear algebra methods such as row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and LU-factorizations are used to confirm the conjectured expressions. The work also supplies closed-form formulas for several related integer sequences that had no prior conjectures.

Core claim

We prove several conjectures recorded in the On-Line Encyclopedia of Integer Sequences. The conjectures considered here concern determinants and permanents of special matrices, such as Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices whose entries are defined by powers of differences. As tools we use row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and LU-factorizations. We also obtain closed-form formulas for several related integer sequences for which no such formulas were conjectured.

What carries the argument

Row and column operations combined with block determinant formulas, Cauchy determinants, Sylvester's theorem, and LU-factorizations applied to the special matrices.

If this is right

The conjectured determinant and permanent values for the Toeplitz and cross matrices are correct.
Determinants of Kronecker powers of the matrices equal the expressions established in the proofs.
Matrices with entries defined by powers of differences have determinants matching the conjectured formulas.
Closed-form expressions are now available for additional related integer sequences.
The same linear algebra techniques confirm the sequences generated by these matrices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The established formulas allow direct evaluation of the sequences for large matrix orders without constructing or factoring the full matrix.
Similar row and block operation methods could be tested on other patterned matrices that appear in OEIS but lack proofs.
The closed forms may reveal combinatorial interpretations or generating-function relations for the underlying sequences.
Extensions to permanents of additional matrix families could follow by adapting the LU-factorization approach.

Load-bearing premise

The matrix definitions and sequence indices stated in the OEIS entries match exactly those used in the proofs.

What would settle it

Direct computation of the determinant or permanent for a specific small matrix size that yields a value different from the conjectured closed form.

read the original abstract

We prove several conjectures recorded in the On-Line Encyclopedia of Integer Sequences. The conjectures considered here concern determinants and permanents of special matrices, such as Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices whose entries are defined by powers of differences. As tools we use row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and $LU$-factorizations. We also obtain closed-form formulas for several related integer sequences for which no such formulas were conjectured.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves several OEIS conjectures on determinants and permanents of Toeplitz, cross, and related matrices using standard linear algebra tools, plus a few new closed forms.

read the letter

The main takeaway is that the author has converted a set of OEIS conjectures into proven closed forms for determinants and permanents of Toeplitz matrices, cross matrices, Kronecker powers, and matrices with entries given by powers of differences. A few additional closed forms appear for sequences that had no prior conjectures.

The proofs rely on row and column operations, block determinant formulas, Cauchy determinants, Sylvester's theorem, and LU factorizations. These are established methods, and the abstract indicates they apply directly to the matrices at hand. The results are new because the formulas were only conjectured before, and the paper supplies the missing derivations.

The work stays narrow. It does not attempt broader generalizations or link the formulas to larger questions in combinatorics, so the contribution remains specific to these sequences. The main setup requirement is that the matrix definitions match the OEIS entries exactly, but once that holds the techniques are deterministic and do not introduce hidden assumptions or regime limits.

Readers who track OEIS entries in matrix enumeration or need explicit formulas for these particular determinants will get direct use from the closed forms. The paper shows straightforward engagement with the listed conjectures and applies the right tools without overclaiming.

It deserves a serious referee. The claims are concrete, the methods are appropriate, and the conjectures were open, so the manuscript should go out for review rather than receive a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper proves several conjectures from the OEIS on the determinants and permanents of special matrices, including Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices with entries defined by powers of differences. Proofs are obtained via row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and LU factorizations. Closed-form formulas are also derived for several related integer sequences where no conjectures had been recorded.

Significance. If the proofs hold, the work resolves multiple open OEIS conjectures in combinatorial matrix theory, supplies explicit closed forms, and demonstrates the applicability of classical linear-algebra tools to these matrix families. The absence of free parameters, ad-hoc constructions, or fitted quantities, together with the use of deterministic named theorems, makes the results reproducible and falsifiable by direct verification against the OEIS entries.

minor comments (2)

The introduction would benefit from an explicit enumerated list of the OEIS entries (A-numbers) treated in each section so that readers can immediately match the proved formulas to the conjectures.
Notation for the matrix families (e.g., the precise definition of the 'cross matrix' and the range of the power-of-difference parameter) should be stated once in a preliminary section rather than repeated inline.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the main results and the classical tools employed.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper provides explicit proofs of OEIS conjectures on determinants and permanents of Toeplitz, cross, Kronecker, and power-of-difference matrices. It relies on standard external tools (row/column operations, block determinant formulas, Cauchy determinants, Sylvester's theorem, LU factorizations) whose application does not reduce to any self-definition, fitted input renamed as prediction, or self-citation load-bearing step. No equations equate a claimed result to its own inputs by construction, and the central claims remain independent of any author-overlapping citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper invokes standard linear-algebra identities (Cauchy determinant, Sylvester theorem, block determinant formulas) whose proofs lie outside this work; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard theorems of linear algebra (Cauchy determinant formula, Sylvester's determinant theorem, block determinant identities) hold.
Abstract lists these as tools; they are background results from prior literature.

pith-pipeline@v0.9.1-grok · 5599 in / 1129 out tokens · 18801 ms · 2026-06-27T19:35:18.692213+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 1 canonical work pages

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work page doi:10.48550/arxiv.2206.12317 2023