Nearly Equal Distributions of the Rank and the Crank of Partitions

Let N(≤ m,n) denote the number of partitions of n with rank not greater than m, and let M(≤ m,n) denote the number of partitions of n with crank not greater than m. Bringmann and Mahlburg observed that N(≤ m,n) ≤ M(≤ m,n) ≤ N(≤ m + 1, n) for m < 0 and 1 ≤ n ≤ 100. They also pointed out that these inequalities can be restated as the existence of a re-ordering τn on the set of partitions of n such that |crank(λ)| − |rank(τn(λ))| = 0 or 1 for all partitions λ of n, that is, the rank and the crank are nearly equal distributions over partitions of n. In the study of the spt-function, Andrews, Dyson and Rhoades proposed a conjecture on the unimodality of the spt-crank, and they showed that this conjecture is equivalent to the inequality N(≤ m,n) ≤ M(≤ m,n) for m < 0 and n ≥ 1. We proved this conjecture by combiantorial arguments. In this paper, we prove the inequality N(≤ m,n) ≤ M(≤ m,n) for m < 0 and n ≥ 1. Furthermore, we define a re-ordering τn of the partitions λ of n and show that this re-ordering τn leads to the nearly equal distribution of the rank and the crank. Using the re-ordering τn, we give a new combinatorial interpretation of the function ospt(n) defined by Andrews, Chan and Kim, which immediately leads to an upper bound for ospt(n) due to Chan and Mao.