Proofs of several OEIS conjectures on determinants and permanents

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.

• The paper provides rigorous proofs of multiple OEIS conjectures, delivering explicit closed-form formulas for determinants and permanents.
• It applies classical matrix techniques like LU decomposition, Cauchy determinants, and Sylvester’s theorem to analyze various structured matrices.
• The work bridges combinatorics and linear algebra, offering insights for future algorithm development and enumerative studies.

Authoritative Summary of "Proofs of several OEIS conjectures on determinants and permanents" (2606.09913)

Overview

This paper rigorously resolves a series of conjectures from the On-Line Encyclopedia of Integer Sequences (OEIS) concerning determinants and permanents of structured matrices. The author systematically addresses conjectures involving Toeplitz, cross, symmetric, Hermitian, arrowhead, and Kronecker product matrices, often providing closed-form expressions for determinant or permanent values. The work heavily leverages classical operators and theorems from matrix theory, including block determinant formulas, Cauchy determinants, Sylvester's theorem, Schur's formula, and LU decompositions. Several integer sequences that previously lacked closed-form expressions also receive explicit formulas.

Methodological Foundation

The author applies a diverse set of linear algebraic techniques:

• Row and Column Operations: Used for structure simplification and to facilitate blockwise decomposition.
• Block Determinant and Schur Formula: Essential for handling multi-block, Toeplitz, and cross matrices.
• Cauchy Determinant and Arrowhead Analysis: Deployed to compute determinants of matrices with polynomially indexed rows/columns.
• Sylvester’s Determinant Theorem: Used notably for Toeplitz-type matrices to reduce computation to low-dimensional auxiliary matrices.
• LU Factorizations: Ascertain closed forms for determinants, especially in the Hermitian Toeplitz context.

The proofs are largely self-contained, relying on established combinatorial and algebraic results besides a handful of references that provide supporting structural theorems.

Resolution and Correction of OEIS Conjectures

Permanent of Floor-Indexed Matrices [A000051]

The paper corrects an OEIS conjecture regarding matrices defined via the floor function and proves that the permanent equals $2^n+1$ only for a specific (corrected) matrix sequence, not as initially conjectured. The counterexample and subsequent valid result highlight the importance of precise indexing and combinatorial interpretation in permanent calculations.

Determinants Related to Dowling Numbers [A007405]

The determinant of a recursively defined matrix is shown to enumerate Dowling numbers, with analytic derivation of the exponential generating function. The recursion is linked via column operations and induction to enumerate $B$-type set partitions and flattened Stirling permutations, confirming the sequence’s combinatorial interpretation.

Cross and Almost-Cross Matrices [A071999]

The determinant formula for an almost-cross matrix is obtained, exploiting the structure of zeros outside main and anti-diagonals. Closed forms are expressed as products over polynomial factors, directly settling the conjecture.

Symmetric Toeplitz and Arrowhead Matrices [A083392, A085799]

For symmetric Toeplitz matrices, a direct relationship is established between their determinant and quadratic floor terms, yielding $(-1)^{n-1}\left\lfloor n^2/4 \right\rfloor$. For matrices indexed by $|i^2-j^2|$, the determinant is derived via block decomposition and arrowhead structure analysis, showing a factorial-polynomial closed form. The Maple counting interpretation is confirmed as equivalent to the matrix determinant's absolute value.

Toeplitz-Type Matrices and Polynomial Families [A323254, A351154]

The determinants of non-standard Toeplitz matrices are computed through recursive column operations and block decomposition, with explicit formulas in terms of $n$ and exponentiation. For matrices defined via the minimum function and linear indexing, the determinant is shown to be simply $-(n-2)!$, demonstrating the power of row-column transformation even in non-classical cases.

Hermitian Toeplitz and Generating Functions [A359559]

A complete LU decomposition is constructed for a family of matrices with imaginary entries, yielding a determinant formula in closed form involving $(1+i)^n$ and $(1-i)^n$ terms. The recurrence relation and generating functions (ordinary and exponential) are derived, confirming OEIS conjectures regarding the nature of these sequences.

Toeplitz-Type Determinants: Faulhaber and Bernoulli Polynomials [A355175, A355326]

Using Sylvester’s determinant theorem and binomial expansions, the author proves that Toeplitz-type determinants parameterized by higher powers have closed forms in terms of Bernoulli polynomials and Faulhaber's formula. The degree and divisibility properties of these polynomials are rigorously established, answering Sun's conjectures about determinant structure.

Binary Matrices and Combinatorial Enumeration [A250742]

A combinatorial analysis leads to a formula for counting binary matrices with prescribed monotonicity properties: $T(n,k) = 2^{n+1} + 2^{k+1} - 2$. The proof elegantly reduces the enumeration to a count of vectors, distinguishing by the initial element and exhaustively tabulating possible intersection cases.

Kronecker Powers and Principal Submatrices [A094384]

Determinants of principal submatrices of Kronecker powers are expressed in terms of the binary digit counts (Hamming weight), showcasing the connection between matrix theory and binary combinatorics. The proof leverages the LU decomposition of the Kronecker base matrix, explicitly tracking diagonal factors through binary expansion.

Numerical Results and Explicit Closed Forms

The paper prominently provides closed-form expressions for all tackled sequences, often in terms of factorial, binomial, or polynomial expressions:

• For cross and Toeplitz families, the determinant formulas typically involve products over structurally indexed terms.
• Permanent formulas are explicitly linked to powers of $2$ or polynomial expressions in $B$0.
• Complex Toeplitz determinants are resolved into linear combinations of $B$1 and $B$2 terms.

The author verifies major explicit formulas algorithmically (Python/Maple), ensuring the correctness beyond symbolic algebra.

Theoretical and Practical Implications

The results deepen the understanding of structured matrices in combinatorics and linear algebra, establishing exact connections between matrix invariants and enumerative sequences. The resolution of OEIS conjectures not only clarifies the combinatorial interpretations but also provides tools for further study in symbolic enumeration, recurrence relations, and generating function characterization.

These proofs enable future researchers to:

• Formulate new conjectures grounded in closed-form and recursive behavior.
• Design fast algorithms for evaluating determinants and permanents of large structured matrices.
• Explore the implications of these formulas in statistical mechanics, discrete probability, coding theory, and theoretical computer science.
• Extend block determinant, LU, and Sylvester’s theorem applications to more generalized matrix families.

Prospects for Future Research

The results open avenues for automated proofs of integer sequence conjectures using advanced symbolic and algorithmic techniques. Insights from determinant structures may inform developments in random matrix theory, spectral analysis of structured graphs, and deep combinatorial enumeration. An explicit characterization of determinants of higher-order Toeplitz and Kronecker product matrices may further bridge combinatorics, number theory, and algebraic geometry.

Conclusion

This paper delivers comprehensive algebraic and combinatorial proofs for multiple OEIS conjectures related to determinants and permanents. Utilizing a toolkit of advanced matrix operations, the author achieves not only elementary closed forms but also validates challenging conjectures involving structured matrices. The work has substantive theoretical value for both combinatorics and linear algebra, and its explicit results facilitate further analytical explorations in discrete mathematics and related fields.