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Andrews and El Bachraoui Conjecture Four

Abstract

The two-color partition series D-prime at parameters 2 and 3 has negative coefficients exactly at degrees 10 and 22.

Theorem 1.1 (The two negative coefficients of D-prime).

Lean statement: D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiFour.result

Proof. Machine-checked in Lean as D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiFour.result (✓ std3). ∎

Resolves. Problems/andrews-el-bachraoui-d23-sign-pattern (proved) by D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiFour.result.

Source. Repository-derived.

Acknowledgement. George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

Commentary.

For every natural number n, the coefficient dCoeff 2 3 n is negative if and only if n is 10 or 22. Thus the two-color partition series D-prime with parameters 2 and 3 has precisely these two negative coefficients. A finite Heine transformation reduces the coefficients to triangular-pair counts and alternating odd divisor sums; their bounds give nonnegativity for large degrees, and exact evaluation treats the remaining degrees.

References