PreprintCC0arXiv
Proof of two conjectures on the log-concavity of partition ranks and cranks
Abstract
Let M(m,n) and N(m,n) denote the numbers of partitions of n with crank m and rank m, respectively. Ji and Zang proved that the crank sequence (M(m,n))|m|≤ n−1 is unimodal for every n≥44. Bringmann, Man, and Rolen subsequently established the unimodality of the rank sequence (N(m,n))|m|≤ n−2 for every n≥39. Ji and Zang also conjectured that (M(m,n))|m|≤ n−71 is log-concave for every n≥72, and that (N(m,n))|m|≤ n−72 is log-concave for every n≥73. In this paper, we prove both conjectures.
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BibTeX
@misc{mao2026proof,
title = {{Proof of two conjectures on the log-concavity of partition ranks and cranks}},
author = {Jianxi Mao and Wenle Shi and Bao-Xuan Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06757},
eprint = {2611.06757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06757},
note = {CC0},
}
RIS
TY - UNPB
TI - Proof of two conjectures on the log-concavity of partition ranks and cranks
AU - Jianxi Mao
AU - Wenle Shi
AU - Bao-Xuan Zhu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06757
UR - https://arxiv.org/abs/2611.06757
AB - Let M(m,n) and N(m,n) denote the numbers of partitions of n with crank m and rank m, respectively. Ji and Zang proved that the crank sequence (M(m,n))_|m|≤ n−1 is unimodal for every n≥44. Bringmann, Man, and Rolen subsequently established the unimodality of the rank sequence (N(m,n))_|m|≤ n−2 for every n≥39. Ji and Zang also conjectured that (M(m,n))_|m|≤ n−71 is log-concave for every n≥72, and that (N(m,n))_|m|≤ n−72 is log-concave for every n≥73. In this paper, we prove both conjectures.
ER -