Asymptotic inequalities for k-ranks and their cumulation functions
Date of Issue2013
School of Physical and Mathematical Sciences
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.
Journal of mathematical analysis and applications
© 2013 Elsevier. This is the author created version of a work that has been peer reviewed and accepted for publication by Journal of Mathematical Analysis and Applications, Elsevier. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [http://dx.doi.org/10.1016/j.jmaa.2013.07.057 ].