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.
| Comments: | We added commentary indicating overlap of arguments here with earlier work by Ballantine et al, and what the new ideas are here |
| Subjects: | Combinatorics (math.CO) |
| MSC classes: | 11R09, 11P81, 05A17, 11A05 |
| Cite as: | arXiv:2605.21718 [math.CO] |
| (or arXiv:2605.21718v2 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.21718 arXiv-issued DOI via DataCite |
From: Ken Ono [view email]
[v1]
Wed, 20 May 2026 20:25:06 UTC (13 KB)
[v2]
Fri, 22 May 2026 04:40:20 UTC (14 KB)
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