Asymptotic inequalities for -ranks and their cumulation functions

Asymptotic formulas for the positive moments of rank and crank of partitions were obtained by K. Bringmann and K. Mahlburg recently. Motivated by their works, in this paper, we prove asymptotic formulas for the k -ranks and their cumulation functions. Asymptotic inequalities between these combinatorial objects are also discovered. In particular, we show that, for fixed integer l and sufficiently large N , M ( l , N ) ∼ N ( l , N ) and M ( l , N ) < N ( l , N ) , where M ( l , N ) (resp. N ( l , N ) ) denotes the number partitions of N with crank (resp. rank) l .