English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 arethe $k$-rank for $k=1$ and $k=2$, respectively. In this paper, we prove thatthe sequence $\{N_k(m,n)\}_{|m|\le n-k-71}$ is log-concave for all sufficientlylarge $n$ and each integer $k$. In particular, we partially solve thelog-concavity conjecture for Andrews-Garvan-Dyson's crank and Dyson's rank,which was independently proposed by Bringmann-Jennings-Shaffer-Mahlburg andJi-Zang.

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