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Andrews and El Bachraoui Conjecture Two

Abstract

The two-color partition series C-prime at parameters 2 and 4 has nonnegative coefficients.

Theorem 1.1 (Positivity of C-prime).

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

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

Resolves. Problems/andrews-el-bachraoui-c24-positivity (proved) by D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiTwo.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 cCoeff 2 4 n is nonnegative. Thus the two-color partition series C-prime with parameters 2 and 4 has no negative coefficient. A finite Heine transformation and partial fractions express the coefficients in terms of divisor counts, a periodic term and weighted alternating sums. Character bounds give nonnegativity for large degrees, and exact evaluation treats the remaining degrees.

References