Some inequalities for Garvan's bicrank function of 2-colored partitions
Volume 190 / 2019
Acta Arithmetica 190 (2019), 171-191
MSC: Primary 05A17; Secondary 11P55, 11P82, 11P83.
DOI: 10.4064/aa180507-7-11
Published online: 28 June 2019
Abstract
In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis’ results on the rank and crank counts for ordinary partitions.
