Partition Function P

P(n), sometimes also denoted p(n) (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 n 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


it follows that P(4)=5. P(n) is sometimes called the number of unrestricted partitions, and is implemented in the Wolfram Language as PartitionsP[n].

The values of P(n) for n=1, 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (OEIS A000041). The values of P(10^n) for n=0, 1, ... are given by 1, 42, 190569292, 24061467864032622473692149727991, ... (OEIS A070177).

The first few prime values of P(n) 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 n giving a probable prime is 1000007396 with 35219 decimal digits (E. Weisstein, Feb. 12, 2017), while the largest known n giving a proven prime is 221444161 with 16569 decimal digits (S. Batalov, Apr. 20, 2017;


When explicitly listing the partitions of a number n, 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 4=2+1+1). The multiplicity representation instead gives the number of times each number occurs together with that number (e.g., (2, 1), (1, 2) for 4=2·1+1·2). The Ferrers diagram is a pictorial representation of a partition. For example, the diagram above illustrates the Ferrers diagram of the partition 6+3+3+2+1=15.

Euler gave a generating function for P(n) using the q-series


Here, the exponents are generalized pentagonal numbers 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (OEIS A001318) and the sign of the kth term (counting 0 as the 0th term) is (-1)^(|_(k+1)/2_|) (with |_x_| the floor function). Then the partition numbers P(n) are given by the generating function


(Hirschhorn 1999).

The number of partitions of a number n into m parts is equal to the number of partitions into parts of which the largest is m, and the number of partitions into at most m parts is equal to the number of partitions into parts which do not exceed m. 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 a_n=1 for all n, then the Euler transform b_n is the number of partitions of n into integer parts.

Euler invented a generating function which gives rise to a recurrence equation in P(n),


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




where sigma_1(n) is the divisor function (Skiena 1990, p. 77; Berndt 1994, p. 108), as well as the identity


where |_x_| is the floor function and [x] is the ceiling function.

A recurrence relation involving the partition function Q is given by


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

sum_(n=0)^(infty)P(5n)q^n=product_(n=1)^(infty)((1-q^(5n-3))(1-q^(5n-2))(1-q^(5n)))/((1-q^(5n-4))^2(1-q^(5n-1))^2) (mod 5)
sum_(n=0)^(infty)P(5n+1)q^n=product_(n=1)^(infty)((1-q^(5n)))/((1-q^(5n-4))(1-q^(5n-1))) (mod 5)
sum_(n=0)^(infty)P(5n+2)q^n=2product_(n=1)^(infty)((1-q^(5n)))/((1-q^(5n-3))(1-q^(5n-2))) (mod 5)
sum_(n=0)^(infty)P(5n+3)q^n=3product_(n=1)^(infty)((1-q^(5n-4))(1-q^(5n-1))(1-q^(5n)))/((1-q^(5n-3))^2(1-q^(5n-2))^2) (mod 5)

(Hirschhorn 1999).

MacMahon obtained the beautiful recurrence relation


where the sum is over generalized pentagonal numbers <=n and the sign of the kth term is (-1)^(|_(k+1)/2_|), as above. Ramanujan stated without proof the remarkable identities


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


(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


(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:




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


where C and D 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 P(n) as a finite sum of algebraic numbers as follows. Define the weight-2 meromorphic modular form F(z) by


were q=e^(2piiz), E_2(q) is an Eisenstein series, and eta(q) is a Dedekind eta function. Now define


where z=x+iy. Additionally let Q_n be any set of representatives of the equivalence classes of the integral binary quadratic form Q(x,y)=ax^2+bxy+cy^2 such that 6|a with a>0 and b=1 (mod 12), and for each Q(x,y), let alpha_Q be the so-called CM point in the upper half-plane, for which Q(alpha_Q,1)=0. Then


where the trace is defined as

 Tr(n)=sum_(Q in Q_n)R(alpha_Q).

Ramanujan found numerous partition function P congruences.

Let f_O(x) be the generating function for the number of partitions P_O(n) of n containing odd numbers only and f_D(x) be the generating function for the number of partitions P_D(n) of n without duplication, then


as discovered by Euler (Honsberger 1985; Andrews 1998, p. 5; Hardy 1999, p. 86), giving the first few values of P_O(n)=P_D(n) for n=0, 1, ... as 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (OEIS A000009). The identity


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



 c_k={(-1)^n   for k of the form 1/2n(3n+/-1); 0   otherwise.

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


(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


The first few values of P^*(n) 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 d times is


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


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


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 n in which no even part is repeated is the same as the number of partitions of n 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 n in which no part occurs more often than d times is the same as the number of partitions in which no term is a multiple of d+1.

3. The number of partitions of n 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 n in which no part appears exactly once is the same as the number of partitions of n 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.

P(n) satisfies the inequality


(Honsberger 1991).

P(n,k) denotes the number of ways of writing n as a sum of exactly k terms or, equivalently, the number of partitions into parts of which the largest is exactly k. (Note that if "exactly k" is changed to "k or fewer" and "largest is exactly k," is changed to "no element greater than k," then the partition function q is obtained.) For example, P(5,3)=2, since the partitions of 5 of length 3 are {3,1,1} and {2,2,1}, and the partitions of 5 with maximum element 3 are {3,2} and {3,1,1}.

The P(n,k) such partitions can be enumerated in the Wolfram Language using IntegerPartitions[n, {k}].

P(n,k) can be computed from the recurrence relation


(Skiena 1990, p. 58; Ruskey) with P(n,k)=0 for k>n, P(n,n)=1, and P(n,0)=0. The triangle of P(k,n) is given by

1  1
1  1  1
1  2  1  1
1  2  2  1  1
1  3  3  2  1  1

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

The recurrence relation can be solved exactly to give


where P(n,k)=0 for n<k. The functions P(n,k) can also be given explicitly for the first few values of k in the simple forms


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


and then yields

P(n,4)={[1/(144)t_4^3(n)-1/(48)t_4(n)] for n even; [1/(144)t_4^3(n)-1/(12)t_4(n)] for n odd.

Hardy and Ramanujan (1918) obtained the exact asymptotic formula


where alpha is a constant. However, the sum


diverges, as first shown by Lehmer (1937).

See also

Alcuin's Sequence, Conjugate Partition, Elder's Theorem, Euler Identity, Ferrers Diagram, Göllnitz's Theorem, Partition, Partition Function P Congruences, Partition Function q, Partition Function Q, Pentagonal Number, Pentagonal Number Theorem, Plane Partition, Random Partition, Rogers-Ramanujan Identities, Self-Conjugate Partition, Stanley's Theorem, Sum of Squares Function, Tau Function

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Partition Function P

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Weisstein, Eric W. "Partition Function P." From MathWorld--A Wolfram Web Resource.

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