Reciprocals of Partition Polynomials
Abstract
Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial attached to an integer partition , 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.
Cite
@article{arxiv.2605.21718,
title = {Reciprocals of Partition Polynomials},
author = {Evan Chen and Ken Ono and Jujian Zhang},
journal= {arXiv preprint arXiv:2605.21718},
year = {2026}
}
Comments
We added commentary indicating overlap of arguments here with earlier work by Ballantine et al, and what the new ideas are here