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Coalescence and the Staircase Pascal Determinant

Abstract

Coalescing alternant variables at one yields a binomial determinant, and a staircase specialization evaluates it.

Theorem 1.1 (The staircase determinant).

Lean statement: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelCoalescence.staircase_det

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PartialTheta/PartialThetaHankelCoalescence.staircase_det (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2024). Hankel determinants of backward shifts of the coefficients of a partial theta function. DOI: 10.48550/arXiv.2407.05768. URL: https://arxiv.org/abs/2407.05768v2.

Commentary.

For all nonnegative integers m and n, form a square integer matrix of size m + n + 1, with row and column indices beginning at zero. In row i below m its entry in column j is one when m - i is at most j and zero otherwise. In row i at least m its entry is binom(j, i - m). The determinant is (-1)^{binom(m + 1, 2)}. Finite differences of the binomial rows give a Pascal determinant of value one, while the staircase rows contribute the stated sign.

Theorem 1.2 (Evaluation of a divided alternant at one).

Lean statement: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelCoalescence.coalescence

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PartialTheta/PartialThetaHankelCoalescence.coalescence (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2024). Hankel determinants of backward shifts of the coefficients of a partial theta function. DOI: 10.48550/arXiv.2407.05768. URL: https://arxiv.org/abs/2407.05768v2.

Commentary.

For nonnegative integers m and c, choose any m by (m + c) matrix T with integer entries and any integer polynomial B in variables x_0 through x_{c-1}. Form the square matrix A of size m + c whose first m rows are T and whose row m + k has entry x_k^j in column j. Suppose det A equals product over 0 at most u less than v below c of (x_v - x_u), multiplied by B. Then B(1, through 1) equals the determinant of the integer matrix with the same first m rows T and with entry binom(j, k) in row m + k and column j. Expansion at x_k = 1 identifies the first alternating homogeneous component with this binomial determinant; the identity also includes c = 0.

References