SOME INEQUALITIES ON RANKS OF CUBIC PARTITIONS

A partition is a cubic partition if its even parts come in two colors (blue and red). Reti defined the rank of a cubic partition as the difference between the number of even parts in blue color and the number of even parts in red color. Motivated by the works on inequalities of rank and crank for certain partitions proved by Andrews and Lewis, and Chern, Fu, Tang and Wang, we prove some inequalities for N′(r,m,n), which count the number of cubic partitions of n whose rank is congruent to r modulo m. More precisely, we establish the generating functions for N′(r,m,n) and determine the signs of the differences N′(r,m,n)−N′(s,m,n) with m∈{2,3,4,6} and 0≤r<s≤m−1 by utilizing q-series techniques in this paper.