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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 Mathematica as PartitionsP[n] or NumberOfPartitions[n] in the Mathematica package Combinatorica`) .

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

The first few prime values of are 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (Sloane's A049575), corresponding to indices 2, 3, 4, 5, 6, 13, 36, 77, 132, ... (Sloane's A046063). The largest known prime value occurs for , has 5254 decimal digits, and was found by P. Minovic in March 2007 (http://primes.utm.edu/primes/page.php?id=79582).

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, ... (Sloane's 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

P(2n+1)=P(n)+sum_(k=1)^infty[P(n-4k^2-3k)+P(n-4k^2+3k)]-sum_(k=1)^infty(-1)^k[P(2n+1-3k^2+k)+P(2n+1-3k^2-k)]

(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:

P(n)=1/(pisqrt(2))sum_(k=1)^inftyA_k(n)sqrt(k)d/(dn)[(sinh(pi/ksqrt(2/3(n-1/(24)))))/(sqrt(n-1/(24)))],

(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).

Ramanujan also 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

			(27)
			(28)
			(29)
			(30)
			(31)
			(32)

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, ... (Sloane's A000009). The identity

(33)

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

			(34)
			(35)

where

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

(36)

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

			(37)
			(38)

(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, ... (Sloane's 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

			(39)
			(40)
			(41)
			(42)

The first few values of are 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, ... (Sloane's 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

			(43)
			(44)

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

			(45)
			(46)
			(47)
			(48)

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, ... (Sloane's A089958; Honsberger 1985, pp. 241-242).

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

			(49)
			(50)
			(51)
			(52)
			(53)

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, ... (Sloane's A007690; Honsberger 1985, p. 241, correcting the sign error in equation 53).

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

(54)

(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 Mathematica using IntegerPartitions[n, k].

can be computed from the recurrence relation

(55)

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

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

(56)

(Sloane's A008284). The number of partitions of with largest part is the same as .

The recurrence relation can be solved exactly to give

			(57)
			(58)
			(59)
		$1/(864){3(n+1)[2n(n+2)-13+9(-1)^n]-96cos(2/3npi)+108(-1)^(n/2)mod(n+1,2)+32sqrt(3)sin(2/3npi)},$	(60)

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

			(61)
			(62)

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

(63)

and then yields

			(64)
			(65)
		${[1/(144)t_4^3(n)-1/(48)t_4(n)] for n even; [1/(144)t_4^3(n)-1/(12)t_4(n)] for n odd.$	(66)

Hardy and Ramanujan (1918) obtained the exact asymptotic formula

(67)

where is a constant. However, the sum

(68)

diverges, as first shown by Lehmer (1937).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Unrestricted Partitions." §24.2.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 825, 1972.

Adler, H. "Partition Identities--From Euler to the Present." Amer. Math. Monthly 76, 733-746, 1969.

Adler, H. "The Use of Generating Functions to Discover and Prove Partition Identities." Two-Year College Math. J. 10, 318-329, 1979.

Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, 1998.

Apostol, T. M. Ch. 4 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.

Apostol, T. M. "Rademacher's Series for the Partition Function." Ch. 5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 94-112, 1997.

Atkin, A. O. L. and Swinnerton-Dyer, P. "Some Properties of Partitions." Proc. London Math. Soc. 4, 84-106, 1954.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 307, 1974.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 94-96, 1996.

Darling, H. B. C. "Proofs of Certain Identities and Congruences Enunciated by S. Ramanujan." Proc. London Math. Soc. 19, 350-372, 1921.

David, F. N.; Kendall, M. G.; and Barton, D. E. Symmetric Function and Allied Tables. Cambridge, England: Cambridge University Press, p. 219, 1966.

Gupta, H. "A Table of Partitions." Proc. London Math. Soc. 39, 142-149, 1935.

Gupta, H. "A Table of Partitions (II)." Proc. London Math. Soc. 42, 546-549, 1937.

Gupta, H.; Gwyther, A. E.; and Miller, J. C. P. Royal Society Mathematical Tables, Vol. 4: Tables of Partitions. London: Cambridge University Press, 1958.

Hardy, G. H. "Ramanujan's Work on Partitions" and "Asymptotic Theory of Partitions." Chs. 6 and 8 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 83-100 and 113-131, 1999.

Hardy, G. H. and Ramanujan, S. "Asymptotic Formulae in Combinatory Analysis." Proc. London Math. Soc. 17, 75-115, 1918.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Hirschhorn, M. D. "Another Short Proof of Ramanujan's Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580-583, 1999.

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 40-45 and 64-68, 1985.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 237-239, 1991.

Jackson, D. and Goulden, I. Combinatorial Enumeration. New York: Academic Press, 1983.

Lehmer, D. H. "On the Hardy-Ramanujan Series for the Partition Function." J. London Math. Soc. 12, 171-176, 1937.

Lehmer, D. H. "On a Conjecture of Ramanujan." J. London Math. Soc. 11, 114-118, 1936.

Lehmer, D. H. "The Series for the Partition Function." Trans. Amer. Math. Soc. 43, 271-295, 1938.

Lehmer, D. H. "On the Remainders and Convergence of the Series for the Partition Function." Trans. Amer. Math. Soc. 46, 362-373, 1939.

MacMahon, P. A. "Note of the Parity of the Number which Enumerates the Partitions of a Number." Proc. Cambridge Philos. Soc. 20, 281-283, 1921.

MacMahon, P. A. "The Parity of , the Number of Partitions of , when ." J. London Math. Soc. 1, 225-226, 1926.

MacMahon, P. A. Combinatory Analysis. New York: Chelsea, 1960.

Mordell, L. J. "Note on Certain Modular Relations Considered by Messrs Ramanujan, Darling and Rogers." Proc. London Math. Soc. 20, 408-416, 1922.

Rademacher, H. "Zur Theorie der Modulfunktionen." J. reine angew. Math. 167, 312-336, 1932.

Rademacher, H. "On the Partition Function ." Proc. London Math. Soc. 43, 241-254, 1937.

Rademacher, H. "On the Expansion of the Partition Function in a Series." Ann. Math. 44, 416-422, 1943.

Ruskey, F. "Information of Numerical Partitions." http://www.theory.csc.uvic.ca/~cos/inf/nump/NumPartition.html.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.

Sloane, N. J. A. Sequences A000009/M0281, A000041/M0663, A000700/M0217, A001318/M1336, A001935/M0566, A007690/M0167, A008284, A046063, A049575, A070177, A089958 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

Uspensky, J. V. "Asymptotic Formulae for Numerical Functions Which Occur in the Theory of Partitions.' Bull. Acad. Sci. URSS 14, 199-218, 1920.

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Weisstein, Eric W. "Partition Function P." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionP.html