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 cranks of partitions. Let M(m,n) denote the number of partitions of n with the Andrews-Garvan-Dyson crank m, we show that the sequence {M(m,n)}|m|≤n−1 is unimodal for n≥44. It turns out that the unimodality of {M(m,n)}|m|≤n−1 is related to the monotonicity properties of two partition functions pr(n) and ppr(n). Let pr(n) denote the number of partitions of n with parts taken from {2,3,…,r} and let ppr(n) denote the number of pairs (α,β) of partitions, where α is a partition counted by pr(i) and β is a partition counted by pr+1(n−i) for 0≤i≤n. We show that pr(n)≥pr(n−1) for r≥5 and n≥14 and ppr(n)≥ppr(n−1) for r≥3 and n≥8. With the aid of the monotonicity properties on pr(n) and ppr(n), we show that M(m,n)≥M(m,n−1) for n≥14 and 0≤m≤n−2 and M(m−1,n)≥M(m,n) for n≥44 and 1≤m≤n−1. By means of the symmetry M(m,n)=M(−m,n), we find that M(m−1,n)≥M(m,n) for n≥44 and 1≤m≤n−1 implies that the sequence {M(m,n)}|m|≤n−1 is unimodal for n≥44. We also give a proof of an upper bound for ospt(n) conjectured by Chan and Mao in light of M(m−1,n)≥M(m,n) for n≥44 and 0≤m≤n−1.