Abstract
AbstractIn 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
}
.
Funder
Max Planck Institute for Mathematics
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference23 articles.
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