Schur's partition theorem states that 
for each nonnegative
integer 
,
where 
counts the partitions of 
into parts congruent to 
(mod 6), 
counts the partitions into distinct parts congruent
to 
(mod 3), and 
counts the partitions into parts that differ by at
least 3, with the added constraint that two multiples of 3 differ by at least 6 (Schur
1926; Bressoud 1980; Andrews 1986, p. 53). All parts are positive
integers, and the empty partition gives 
.
The values of 
for 
,
2, ... are 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, ... (OEIS
A003105). For example, for 
, there are nine partitions
satisfying these conditions, as summarized in the following table (Andrews 1986,
p. 54).
 |  |  |
 |  | 15 |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
The identity 
can be established using the generating
function identity
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
for 
(Andrews 1986, p. 54). Here 
is the q-Pochhammer
symbol. The identity 
is significantly trickier.
(mod 
)." Jahresber. Deutsche Math.-Verein. 25,
114-116, 1916.Schur, I. "Zur additiven Zahlentheorie." Sitzungsber.
Preuss. Akad. Wiss. Phys.-Math. Kl., pp. 488-495, 1926. Reprinted in Gesammelte
Abhandlungen, Vol. 3. Berlin, Germany: Springer-Verlag, pp. 43-50,
1973.Sloane, N. J. A. Sequence