Some inequalities for Garvan's bicrank function of 2-colored partitions

TL;DR

This paper establishes inequalities among bicrank counts for 2-colored partitions using asymptotic analysis and q-series techniques, extending results similar to those for ordinary partition ranks and cranks.

Contribution

It introduces new inequalities for Garvan's bicrank counts of 2-colored partitions, providing a unified combinatorial interpretation of certain congruences.

Findings

Inequalities between bicrank counts for m=2, 3, 4 derived

Asymptotic formulas used to establish inequalities

Results parallel known inequalities for ordinary partition ranks and cranks

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.