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 modulo $$ c\ge 2. $$ c ≥ 2 . If $$ \overline{N}(a,c,n) $$ N ¯ ( a , c , n ) denotes the number of overpartitions of n with rank congruent to a modulo c, we prove that for any $$ c\ge 7 $$ c ≥ 7 and $$ 0\le a<b\le \left\lfloor \frac{c}{2}\right\rfloor $$ 0 ≤ a < b ≤ c 2 we have $$ \overline{N}(a,c,n)>\overline{N}(b,c,n) $$ N ¯ ( a , c , n ) > N ¯ ( b , c , n ) for n large enough. That the sign of the rank differences $$ \overline{N}(a,c,n)-\overline{N}(b,c,n) $$ 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 $$ c=6, $$ 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 $$ c\in \{2,3, 4,5\}. $$ c ∈ { 2 , 3 , 4 , 5 } .