Let 
denote the number of partitions of 
into parts 
(mod 12), let 
denote the number of partitions
of 
into distinct parts 
(mod 6), and let 
denote the number of partitions of 
of the form
 |
(1)
|
where 
,
with strict inequality if 
or 3 (mod 6), and 
. Then
 |
(2)
|
(Andrews 1986, p. 101).
The values of 
for 
,
2, ... are 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8,
9, ... (OEIS A056970). For example, for 
, there are eight partitions satisfying
these conditions, as summarized in the following table.
 |  |  |
 |  | 24 |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
The identity 
can be established using the identity
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
where 
is a q-Pochhammer symbol (Andrews 1986,
p. 101). The assertion 
is significantly more difficult, and no simple proof
is known. However, it can be established with the aid of computer algebra and the
following refinement of the Göllnitz theorem.
Let 
denote the number of partitions of 
into 
distinct parts 
, 4, 5 (mod 6). Let 
denote the number of partitions of 
of the form
 |
(10)
|
where 
,
with strict inequality if 
, 1, 3 (mod 6), where 
, 3, and 
is the number of 
plus twice the number of 
. Then 
for each 
and 
(Göllnitz 1967; Andrews 1986, p. 102).
