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

Abstract

Let Nk(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 sequences (Nk(m,n))|m|≤n−k−71 are log-concave for all sufficiently large integers 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 recently.