  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.
