Unimodality of the Andrews-Garvan-Dyson cranks of partitions

Abstract

The main objective of this paper is to investigate the distribution of the Andrews-Garvan-Dyson crank of a partition. Let $M(m,n)$ denote the number of partitions of $n$ with the Andrews-Garvan-Dyson crank $m$, we show that the sequence \break $\{M(m,n){\}}_{|m|\le n-1}$ is unimodal for $n\ge 44$. It turns out that the unimodality of \break $\{M(m,n){\}}_{|m|\le n-1}$ is related to the monotonicity properties of two partition \break functions $p_k(n)$ and $p{p}_k(n)$. Let $p_k(n)$ denote the number of partitions of $n$ with at most $k$ parts such that the largest part appears at least twice and let $p{p}_k(n)$ denote the number of pairs $(\alpha ,\beta )$ of partitions of $n$, where $\alpha$ is a partition counted by $p_k(i)$ and $\beta$ is a partition counted by $p_{k+1}(n-i)$ for $0\le i\le n$. We show that $p_k(n)\ge p_k(n-1)$ for $k\ge 5$ and $n\ge 14$ and $p{p}_k(n)\ge p{p}_k(n-1)$ for $k\ge 3$ and $n\ge 2$. With the aid of the monotonicity properties on $p_k(n)$ and $p{p}_k(n)$, we show that $M(m,n)\ge M(m,n-1)$ for $n\ge 14$ and $0\le m\le n-2$ and $M(m-1,n)\ge M(m,n)$ for $n\ge 44$ and $1\le m\le n-1$. By means of the symmetry $M(m,n)=M(-m,n)$, we find that $M(m-1,n)\ge M(m,n)$ for $n\ge 44$ and $1\le m\le n-1$ implies that the sequence $\{M(m,n){\}}_{|m|\le n-1}$ is unimodal for $n\ge 44$. We also give a proof of an upper bound for ospt(n) conjectured by Chan and Mao in light of the inequality $M(m-1,n)\ge M(m,n)$ for $n\ge 44$ and $0\le m\le n-1$.

53 pages, 1 figure, to appear in Adv. in Math

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