Unimodality of the Andrews-Garvan-Dyson cranks of partitions

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