Asymptotic inequalities for k-ranks and their cumulation functions - Mao, Renrong (2014)
Asymptotic formulas for the positive moments of rank and crank of partitions were obtained by K. Bringmann and K. Mahlburg recently. Mo-tivated 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 su ciently large N, M(l;N) N(l;N) and M(l;N) < N(l;N); whereM(l;N) (resp. N(l;N)) denotes the number partitions of N with crank (resp. rank) l.