Partition Function P

, sometimes also denoted  (Abramowitz and Stegun 1972, p. 825; Comtet 1974, p. 94; Hardy and Wright 1979, p. 273; Conway and Guy 1996, p. 94; Andrews 1998, p. 1), gives the number of ways of writing the integer  as a sum of positive integers, where the order of addends is not considered significant. By convention, partitions are usually ordered from largest to smallest (Skiena 1990, p. 51). For example, since 4 can be written

			(1)
			(2)
			(3)
			(4)
			(5)

it follows that .  is sometimes called the number of unrestricted partitions, and is implemented in the Wolfram Language as PartitionsP[n].

The values of  for , 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (OEIS A000041). The values of  for , 1, ... are given by 1, 42, 190569292, 24061467864032622473692149727991, ... (OEIS A070177).

The first few prime values of  are 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575), corresponding to indices 2, 3, 4, 5, 6, 13, 36, 77, 132, ... (OEIS A046063). As of Feb. 3, 2017, the largest known  giving a probable prime is  with  decimal digits (E. Weisstein, Feb. 12, 2017), while the largest known  giving a proven prime is  with  decimal digits (S. Batalov, Apr. 20, 2017; https://primes.utm.edu/top20/page.php?id=54#records).

When explicitly listing the partitions of a number , the simplest form is the so-called natural representation which simply gives the sequence of numbers in the representation (e.g., (2, 1, 1) for the number ). The multiplicity representation instead gives the number of times each number occurs together with that number (e.g., (2, 1), (1, 2) for ). The Ferrers diagram is a pictorial representation of a partition. For example, the diagram above illustrates the Ferrers diagram of the partition .

Euler gave a generating function for  using the q-series

			(6)
			(7)
			(8)

Here, the exponents are generalized pentagonal numbers 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (OEIS A001318) and the sign of the th term (counting 0 as the 0th term) is  (with  the floor function). Then the partition numbers  are given by the generating function

			(9)
			(10)

(Hirschhorn 1999).

The number of partitions of a number  into  parts is equal to the number of partitions into parts of which the largest is , and the number of partitions into at most  parts is equal to the number of partitions into parts which do not exceed . Both these results follow immediately from noting that a Ferrers diagram can be read either row-wise or column-wise (although the default order is row-wise; Hardy 1999, p. 83).

For example, if  for all , then the Euler transform  is the number of partitions of  into integer parts.

Euler invented a generating function which gives rise to a recurrence equation in ,

(11)

(Skiena 1990, p. 57). Other recurrence equations include

(12)

and

(13)

where  is the divisor function (Skiena 1990, p. 77; Berndt 1994, p. 108), as well as the identity

(14)

where  is the floor function and  is the ceiling function.

A recurrence relation involving the partition function Q is given by

(15)

Atkin and Swinnerton-Dyer (1954) obtained the unexpected identities

			(16)
			(17)
			(18)
			(19)

(Hirschhorn 1999).

MacMahon obtained the beautiful recurrence relation

(20)

where the sum is over generalized pentagonal numbers  and the sign of the th term is , as above. Ramanujan stated without proof the remarkable identities

(21)

(Darling 1921; Mordell 1922; Hardy 1999, pp. 89-90), and

(22)

(Mordell 1922; Hardy 1999, pp. 89-90, typo corrected).

Hardy and Ramanujan (1918) used the circle method and modular functions to obtain the asymptotic solution

(23)

(Hardy 1999, p. 116), which was also independently discovered by Uspensky (1920). Rademacher (1937) subsequently obtained an exact convergent series solution which yields the Hardy-Ramanujan formula (23) as the first term:

(24)

where

(25)

 is the Kronecker delta, and  is the floor function (Hardy 1999, pp. 120-121). The remainder after  terms is

(26)

where  and  are fixed constants (Apostol 1997, pp. 104-110; Hardy 1999, pp. 121 and 128). Rather amazingly, the contour used by Rademacher involves Farey sequences and Ford circles (Apostol 1997, pp. 102-104; Hardy 1999, pp. 121-122). In 1942, Erdős showed that the formula of Hardy and Ramanujan could be derived by elementary means (Hoffman 1998, p. 91).

Bruinier and Ono (2011) found an algebraic formula for the partition function  as a finite sum of algebraic numbers as follows. Define the weight-2 meromorphic modular form  by

(27)

were ,  is an Eisenstein series, and  is a Dedekind eta function. Now define

(28)

where . Additionally let  be any set of representatives of the equivalence classes of the integral binary quadratic form  such that  with  and , and for each , let  be the so-called CM point in the upper half-plane, for which . Then

(29)

where the trace is defined as

(30)

Ramanujan found numerous partition function P congruences.

Let  be the generating function for the number of partitions  of  containing odd numbers only and  be the generating function for the number of partitions  of  without duplication, then

			(31)
			(32)
			(33)
			(34)
			(35)
			(36)

as discovered by Euler (Honsberger 1985; Andrews 1998, p. 5; Hardy 1999, p. 86), giving the first few values of  for , 1, ... as 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (OEIS A000009). The identity

(37)

is known as the Euler identity (Hardy 1999, p. 84).

The generating function for the difference between the number of partitions into an even number of unequal parts and the number of partitions in an odd number of unequal parts is given by

			(38)
			(39)

where

(40)

Let  be the number of partitions of even numbers only, and let  () be the number of partitions in which the parts are all even (odd) and all different. Then the generating function of  is given by

			(41)
			(42)

(Hardy 1999, p. 86), and the first few values of are 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, ... (OEIS A000700). Additional generating functions are given by Honsberger (1985, pp. 241-242).

Amazingly, the number of partitions with no even part repeated is the same as the number in which no part occurs more than three times and the same as the number in which no part is divisible by 4, all of which share the generating functions

			(43)
			(44)
			(45)
			(46)

The first few values of  are 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, ... (OEIS A001935; Honsberger 1985, pp. 241-242).

In general, the generating function for the number of partitions in which no part occurs more than  times is

			(47)
			(48)

(Honsberger 1985, pp. 241-242). The generating function for the number of partitions in which every part occurs 2, 3, or 5 times is

			(49)
			(50)
			(51)
			(52)

The first few values are 0, 1, 1, 1, 1, 3, 1, 3, 4, 4, 4, 8, 5, 9, 11, 11, 12, 20, 15, 23, ... (OEIS A089958; Honsberger 1985, pp. 241-242).

The number of partitions in which no part occurs exactly once is

			(53)
			(54)
			(55)
			(56)
			(57)

The first few values are, 1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 16, 11, 22, 20, 33, 28, 51, 42, 71, ... (OEIS A007690; Honsberger 1985, p. 241, correcting the sign error in equation 57).

Some additional interesting theorems following from these (Honsberger 1985, pp. 64-68 and 143-146) are:

1. The number of partitions of  in which no even part is repeated is the same as the number of partitions of  in which no part occurs more than three times and also the same as the number of partitions in which no part is divisible by four.

2. The number of partitions of  in which no part occurs more often than  times is the same as the number of partitions in which no term is a multiple of .

3. The number of partitions of  in which each part appears either 2, 3, or 5 times is the same as the number of partitions in which each part is congruent mod 12 to either 2, 3, 6, 9, or 10.

4. The number of partitions of  in which no part appears exactly once is the same as the number of partitions of  in which no part is congruent to 1 or 5 mod 6.

5. The number of partitions in which the parts are all even and different is equal to the absolute difference of the number of partitions with odd and even parts.

 satisfies the inequality

(58)

(Honsberger 1991).

 denotes the number of ways of writing  as a sum of exactly  terms or, equivalently, the number of partitions into parts of which the largest is exactly . (Note that if "exactly " is changed to " or fewer" and "largest is exactly ," is changed to "no element greater than ," then the partition function q is obtained.) For example, , since the partitions of 5 of length 3 are  and , and the partitions of 5 with maximum element 3 are  and .

The  such partitions can be enumerated in the Wolfram Language using IntegerPartitions[n, k].

 can be computed from the recurrence relation

(59)

(Skiena 1990, p. 58; Ruskey) with  for , , and . The triangle of  is given by

(60)

(OEIS A008284). The number of partitions of  with largest part  is the same as .

The recurrence relation can be solved exactly to give

			(61)
			(62)
			(63)
			(64)

where  for . The functions  can also be given explicitly for the first few values of  in the simple forms

			(65)
			(66)

where  is the floor function and  is the nearest integer function (Honsberger 1985, pp. 40-45). A similar treatment by B. Schwennicke defines

(67)

and then yields

			(68)
			(69)
			(70)

Hardy and Ramanujan (1918) obtained the exact asymptotic formula

(71)

where  is a constant. However, the sum

(72)

diverges, as first shown by Lehmer (1937).

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