Inequalities between overpartition ranks for all moduli

In this paper we give a full description of the inequalities that can occur between overpartition ranks modulo c ≥ 2 . If N ¯ ( a , c , n ) denotes the number of overpartitions of n with rank congruent to a modulo c , we prove that for any c ≥ 7 and 0 ≤ a < b ≤ c 2 we have N ¯ ( a , c , n ) > N ¯ ( b , c , n ) for n large enough. That the sign of the rank differences N ¯ ( a , c , n ) - N ¯ ( b , c , n ) depends on the residue class of n modulo c in the case of small moduli, such as c = 6 , is known due to the work of Ji et al. (J Number Theory 184:235–269, 2018) and Ciolan (Int J Number Theory 16(1):121–143, 2020). We show that the same behavior holds for c ∈ { 2 , 3 , 4 , 5 } .