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Proof of Almkvist's conjecture on the unimodality of partition polynomials

  • Jianxi Mao
  • Wenle Shi
  • Bao-xuan Zhu
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Abstract

For integers r≥2 and n≥1, let Fr,n(q)=∏k=1n(1−qrk)/(1−qk). The coefficient of qj in Fr,n(q) counts partitions of j into parts at most n, each occurring at most r−1 times. Hughes proved that F2,n(q)=∏k=1n(1+qk) is unimodal for every n≥ 1. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that Fr,n(q) is unimodal in the following two cases: every even r and every n≥ 1; every odd r and every n≥ 11. He proved that this conjecture is true for 3≤ r ≤ 20 and r=100,101. In this paper, we completely settle the conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{mao2026proof,
  title = {{Proof of Almkvist's conjecture on the unimodality of partition polynomials}},
  author = {Jianxi Mao and Wenle Shi and Bao-xuan Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.23779},
  eprint = {2609.23779},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23779},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Proof of Almkvist's conjecture on the unimodality of partition polynomials
AU  - Jianxi Mao
AU  - Wenle Shi
AU  - Bao-xuan Zhu
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.23779
UR  - https://arxiv.org/abs/2609.23779
AB  - For integers r≥2 and n≥1, let F_r,n(q)=∏_k=1^n(1−q^rk)/(1−q^k). The coefficient of q^j in F_r,n(q) counts partitions of j into parts at most n, each occurring at most r−1 times. Hughes proved that F_2,n(q)=∏_k=1^n(1+q^k) is unimodal for every n≥ 1. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that F_r,n(q) is unimodal in the following two cases: every even r and every n≥ 1; every odd r and every n≥ 11. He proved that this conjecture is true for 3≤ r ≤ 20 and r=100,101. In this paper, we completely settle the conjecture.
ER  -