Eventual log-concavity of k-rank statistics for integer partitions

2021/10/21 by Nian Hong Zhou, Zhou, Nian Hong
Mathematics · #05A17 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11P82 #Random Matrices and Applications #Secondary 05A16

paper · pdf · doi:10.48550/arxiv.2110.11174

openalex publication_date 2021/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let N_k(m,n) denote the number of partitions of n with Garvan k-rank m. It is well-known that Andrews-Garvan-Dyson's crank and Dyson's rank are the k-rank for k=1 and k=2, respectively. In this paper, we prove that the sequence \N_k(m,n)\_{|m|≤ n-k-71} is log-concave for all sufficiently large n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews-Garvan-Dyson's crank and Dyson's rank, which was independently proposed by Bringmann-Jennings-Shaffer-Mahlburg and Ji-Zang.

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