Reciprocals of Partition Polynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914589896081408 |
|---|---|
| author | Chen, Evan Ono, Ken Zhang, Jujian |
| author_facet | Chen, Evan Ono, Ken Zhang, Jujian |
| contents | Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(λ,x):=\prod_i (1+x^{λ_i}) \] attached to an integer partition $λ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21718 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reciprocals of Partition Polynomials Chen, Evan Ono, Ken Zhang, Jujian Combinatorics 11R09, 11P81, 05A17, 11A05 Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(λ,x):=\prod_i (1+x^{λ_i}) \] attached to an integer partition $λ$, 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. |
| title | Reciprocals of Partition Polynomials |
| topic | Combinatorics 11R09, 11P81, 05A17, 11A05 |
| url | https://arxiv.org/abs/2605.21718 |