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

TL;DR

This paper proves that for large enough n, the sequence of k-rank partition counts is log-concave, advancing understanding of partition statistics and partially confirming a longstanding conjecture for specific cases.

Contribution

It establishes the log-concavity of k-rank partition counts for large n, partially resolving a conjecture related to the Andrews-Garvan-Dyson crank and Dyson's rank.

Findings

Sequence {N_k(m,n)} is log-concave for large n.

Partially confirms the log-concavity conjecture for specific partition statistics.

Advances understanding of the distribution of k-rank partition counts.

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 $\{N_k(m,n)\}_{|m|\le n-k-71}$ 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.