Inequalities between overpartition ranks for all moduli
Abstract
In this paper we give a full description of the inequalities that can occur between overpartition ranks. If N(a,c,n) denotes the number of overpartitions of n with rank congruent to a modulo c, we prove that for any c7 and 0 a<bc2 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, Zhang and Zhao (2018) and Ciolan (2020). We show that the same behavior holds for c∈\2,3, 4,5\.
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