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 qPochhammer 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).