The partition function 
gives the number of partitions
of 
with 
or
fewer addends, or equivalently, into partitions with
no element greater than 
. This function is denoted 
or 
. (Note that if "
or fewer" is changed to "exactly 
" and "no element greater than 
" to "greatest element equal to 
," then the partition
function P of two arguments is obtained.)
The 
such partitions can be enumerated in the Wolfram
Language using IntegerPartitions[n,
k].
For example, the 
partitions of 5 of which the largest member is 
are 
, 
, 
, 
, and 
. Similarly, the five partitions of 5 into three
or fewer parts are 
,
,
,
,
and 
.
The 
satisfy the recurrence relation
 |
(1)
|
with 
,
,
and 
for 
.
The triangle of 
is given by
 |
(2)
|
(OEIS A026820).
The partition function 
, also denoted 
(Abramowitz and Stegun 1972, p. 825), gives the number
of ways of writing the integer 
as a sum of positive integers
without regard to order with the constraint that all integers
in a given partition are distinct. For example, 
, since the partitions of 10 into distinct parts are
,
,
,
,
,
,
,
,
,
.
The 
function is implemented in the Wolfram
Language as PartitionsQ[n].
is generally defined to be 1.
The values for 
,
2, ... are 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (OEIS A000009).
The first few prime values of 
are for indices 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072,
320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852, ... (OEIS A035359),
corresponding to values 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (OEIS
A051005), with no others up to 
(M. Alekseyev, Jul. 10, 2015).
is also the number of partitions of 
with odd parts, sometimes denoted 
(Andrews 1998, p. 237).
The generating function for 
is
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
and 
areq-Pochhammer symbols.
This can also be interpreted as another form of the Jacobi triple product, written in terms of the Q-functions as
 |
(9)
|
(Borwein and Borwein 1987, p. 64).
A recurrence relation is given by 
and
![Q(n)=1/nsum_(k=1)^n[s(k)-2s(1/2k)]Q(n-k),](/images/equations/PartitionFunctionQ/NumberedEquation4.svg) |
(10)
|
where
 |
(11)
|
and
 |
(12)
|
is the odd divisor function giving the sum of odd divisors of 
:
1, 1, 4, 1, 6, 4, 8, ... (OEIS A000593; Abramowitz
and Stegun 1972, p. 826).
Another recurrence relation is given by 
and
 |
(13)
|
where
 |
(14)
|
(E. Georgiadis, A. V. Sutherland, and K. S. Kedlaya; Sloane).
satisfies the inequality
![Q(n)<=1/2[Q(n+1)+Q(n-1)]](/images/equations/PartitionFunctionQ/NumberedEquation9.svg) |
(15)
|
for 
.
has the asymptotic series
 |
(16)
|
(Abramowitz and Stegun 1972, p. 826).
A Rademacher-like convergent series for 
is given by
![Q(n)=1/2sqrt(2)sum_(k=1)^inftyA_(2k-1)(n){d/(dn^')[J_0((pii)/(2k-1),sqrt(1/3(n^'+1/(24))))]}_(n^'=n),](/images/equations/PartitionFunctionQ/NumberedEquation11.svg) |
(17)
|
where
![A_k(n)=sum_(h=1; (h,k)=1)^ke^(pii[s(h,k)-s(2h,k)])e^(-2piihn/k)](/images/equations/PartitionFunctionQ/NumberedEquation12.svg) |
(18)
|
(P. J. Grabner, pers. comm., Sep. 10, 2003; Hagis 1964ab, 1965), where 
means 
and 
are relatively prime,
 |
(19)
|
is a Dedekind sum, 
is the floor function,
and 
is the zeroth order Bessel function
of the first kind. Equation (18) corrects Abramowitz and
Stegun (1972, p. 825), which erroneously state to be identical to the analogous
expression in the formula for 
). (17) can also be written explicitly
as
 |
(20)
|
where 
is a generalized hypergeometric
function.
Let 
denote the number of ways of partitioning 
into exactly 
distinct parts. For example, 
since there are four partitions of 10 into three distinct
parts: 
,
,
,
and 
.
is given by
 |
(21)
|
where 
is the partition function P and 
is a binomial coefficient
(Comtet 1974, p. 116). The following table gives the first few values of 
(OEIS A008289;
Comtet 1974, pp. 115-116).
 | 1 | 2 | 3 | 4 |
| 1 | 1 | |||
| 2 | 1 | |||
| 3 | 1 | 1 | ||
| 4 | 1 | 1 | ||
| 5 | 1 | 2 | ||
| 6 | 1 | 2 | 1 | |
| 7 | 1 | 3 | 1 | |
| 8 | 1 | 3 | 2 | |
| 9 | 1 | 4 | 3 | |
| 10 | 1 | 4 | 4 | 1 |
