English

Reciprocals of Partition Polynomials

Combinatorics 2026-05-25 v2

Abstract

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial sp(λ,x):=i(1+xλi) \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) attached to an integer partition λ\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.

Keywords

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

R2 v1 2026-07-22T07:24:55.546Z