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arXiv · 2605.21718

Reciprocals of Partition Polynomials

Evan Chen·Ken Ono·Jujian Zhang
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Abstract

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals of these polynomials over natural classes of partitions. They posed ten conjectures which naturally divide into coprimality and divisibility questions, special-value and recurrence formulas, and coefficient-shape problems. We prove all of the conjectures in the first two families: the ordinary and binary coprimality/divisibility conjectures, and the odd and ternary special-value/recurrence conjectures. AxiomProver autonomously produced Lean/mathlib formalizations and machine-checkable proofs of these six conjectures, and also discovered a counterexample to the statement as printed; the corrected form remains open.

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Evan Chen, Ken Ono, Jujian Zhang. 2026-05-20. Reciprocals of Partition Polynomials. https://arxiv.org/abs/2605.21718

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