A remark on tail distributions of partition rank and crank (Q2786574)

In order to combinatorially explain the famous Ramanujan congruences mod\,5,7 and 11 for the number \(p(n)\) of the partitions of \(n\), \textit{F. Dysen} [Eureka 8, 10--15 (1944)] defined the rank of a partition as the largest part minus the number of parts; \textit{G. E. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167--171 (1988; Zbl 0646.10008)] defined the crank of a partition as the largest part if the partition contains no ones, and otherwise the number of parts larger than the number of ones minus the number of ones. Let \(M(m,n)\) (resp. \(N(m,n)\)) denote the number of partitions of \(n\) with crank (resp. rank) \(m\). \textit{A. O. L. Atkin} and \textit{F. G. Garvan} [Ramanujan J. 7, No. 1--3, 343--366 (2003; Zbl 1039.11069)] introduced the moments NEWLINE\[NEWLINE\sum_{m\in\mathbb{Z}} m^kM(m,n)\quad\text{and}\quad \sum_{m\in\mathbb{Z}} m^kN(m,n).NEWLINE\]NEWLINE Summing only for positive \(m\)'s, \textit{G. E. Andres} et al. [J. Comb. Theory, Ser. A 120, No. 1, 77--91 (2013; Zbl 1264.11088)] introduced the positive moments and proved that, for positive integers \(k\) and \(n\), the \(k\)th positive moment of the crank of \(n\) is strictly larger than the \(k\)th positive moment of the rank of \(n\).NEWLINENEWLINE In the paper under review, for a positive integer \(r\), the author introduces the tail moments with summations only over \(m\geq r\) and proves the analogous inequality for \(k=1\) and an asymptotic relation for the difference.