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In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. For example, 4 can be partitioned in 5 distinct ways: Wikipedia does not have an article with this exact name. ...
In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...
Number theory is the formal study of numbers. ...
Number is the current mathematics collaboration of the week! Please help improve it to featured article standard. ...
In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...
A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
Addition is one of the basic operations of arithmetic. ...
- 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.
So p(4) = 5. By convention p(0) = 1, p(n) = 0 for n negative. Partitions can be graphically visualized with Young diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general. In mathematics, a Young tableau is a combinatorial object useful in representation theory. ...
Euclid, detail from The School of Athens by Raphael. ...
A Superconductor demonstrating the Meissner Effect. ...
In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ...
In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...
Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
Intermediate function
One way of getting a handle on the partition function involves an intermediate function p(k, n) which represents the number of partitions of n using only natural numbers at least as large as k. For any given value of k, partitions counted by p(k, n) fit into exactly one of the following categories: - smallest addend is k
- smallest addend is strictly greater than k
The number of partitions meeting the first condition is p(k, n − k). To see this, imagine a list of all the partitions of the number n − k into numbers of size at least k, then imagine appending "+ k" to each partition in the list. Now what is it a list of? Addition is one of the basic operations of arithmetic. ...
The number of partitions meeting the second condition is p(k + 1, n) since a partition into parts of at least k which contains no parts of exactly k must have all parts at least k + 1.
Since the two conditions are mutually exclusive, the number of partitions meeting either condition is p(k + 1, n) + p(k, n − k). The base cases of this recursively defined function are as follows: In logic, two mutually exclusive (or mutual exclusive according to some sources) propositions are propositions that logically cannot both be true. ...
A Sierpinski triangle —a confined recursion of triangles to form a geometric lattice. ...
This function tends to exhibit deceptive behavior. - p(1, 4) = 5
- p(2, 8) = 7
- p(3, 12) = 9
- p(4, 16) = 11
- p(5, 20) = 13
- p(6, 24) = 16
Our original function p(n) is just p(1, n).
The values of this function:
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| k | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | n | 1 | 1 | | 2 | 2 | 1 | | 3 | 3 | 1 | 1 | | 4 | 5 | 2 | 1 | 1 | | 5 | 7 | 2 | 1 | 1 | 1 | | 6 | 11 | 4 | 2 | 1 | 1 | 1 | | 7 | 15 | 4 | 2 | 1 | 1 | 1 | 1 | | 8 | 22 | 7 | 3 | 2 | 1 | 1 | 1 | 1 | | 9 | 30 | 8 | 4 | 2 | 1 | 1 | 1 | 1 | 1 | | 10 | 42 | 12 | 5 | 3 | 2 | 1 | 1 | 1 | 1 | 1 |
Generating function A generating function for p(n) is given by the reciprocal of Euler's function: In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...
In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...
Modulus of phi on the complex plane, colored so that black=0, red=4 For other meanings, see Euler function (disambiguation). ...
 Expanding each term on the right-hand side as a geometric series, we can rewrite it as In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...
- (1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) ...
The xn term in this product counts the number of ways to write - n = a1 + 2a2 + 3a3 + ... = (1 + 1 + ... + 1) + (2 + 2 + ... + 2) + (3 + 3 + ... + 3) + ...,
where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the desired generating function. More generally, the generating function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler. It has been suggested that Leonhard Euler/EB1911 biography be merged into this article or section. ...
The formulation of Euler's generating function is a special case of a q-series and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function. It can also be used in conjunction with the pentagonal number theorem to derive a recurrence for the partition function stating that In mathematics, a q-series, also sometimes called a q-shifted factorial, is defined as It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ...
Modular form - Wikipedia /**/ @import /skins-1. ...
The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...
In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. ...
- p(k) − p(k − 1) − p(k − 2) + p(k − 5) + p(k − 7) − p(k − 12) − ... = 0,
where the sum is taken over all pentagonal numbers of the form ½n(3n − 1), including those where n < 0, and the terms continue to alternate +, +, −, −, +, +, ... In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ...
Table of values Some values of the partition function are as follows: - p(1) = 1
- p(2) = 2
- p(3) = 3
- p(4) = 5
- p(5) = 7
- p(6) = 11
- p(7) = 15
- p(8) = 22
- p(9) = 30
- p(10) = 42
- p(100) = 190569292
- p(1000) = 24061467864032622473692149727991
Rademacher's series An asymptotic expression for p(n) is given by In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...
 This expression was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...
1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ...
James Victor Uspensky (1883 - 1947) was a mathematician who wrote Theory of Equations. ...
1920 (MCMXX) was a leap year starting on Thursday (link will take you to calendar) // Events January January 7 - Forces of Russian White admiral Kolchak surrender in Krasnoyarsk. ...
In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ...
Hans Rademacher (1892 - 1969) was a German mathematician, known for work in mathematical analysis and number theory. ...
In mathematics, a series is the sum of the terms of a sequence of numbers. ...
 where  Here, the notation (m,n) = 1 implies that the sum should occur only over the values of m that are relatively prime to n. The function s(m,k) is a Dedekind sum. The proof of Rademacher's formula is interesting in that it involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way. In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
In mathematics, Dedekind sums are certain sums of products of a sawtooth function s, and are given by a function D of three integer variables. ...
In mathematics a Ford circle is a circle with centre at (p/q, 1/2q2) and radius 1/(2q2), where p/q is a fraction in its lowest terms (i. ...
In mathematics, a Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. ...
In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...
Congruences Mathematician Srinivasa Ramanujan is credited with discovering that "congruences" in the number of partitions exist for integers ending in 4 and 9. Starting at the number 4, the partition number for every 5th integer is divisible by 5. For instance, the number of partitions for the integer 4 is 5. For the integer 9, the number of partitions is 30; for 14 there are 135 partitions. The integers 5, 30, and 135 are all evenly divisible by 5. Ramanujan demonstrated that this congruence goes indefinitely. He also discovered congruences related to 7 and 11. Starting with 5, the partition number for every 7th integer (5, 12, 19, ... ) is itself divisible by 7. Starting with 6, the partition number for every 11th integer is divisible by 11. Since 5, 7, and 11 are consecutive primes, one might think that starting with 7, the partition number of every 13th integer is divisible by 13. This is, however, false. This article is in need of attention from an expert on the subject. ...
Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸà¯à®°à¯€à®©à®¿à®µà®¾à®¸ à®à®¯à®™à¯à®•à®¾à®°à¯ ராமானà¯à®œà®©à¯) (December 22, 1887 – April 26, 1920) was an Indian mathematician and one of the most esoteric mathematical geniuses in the twentieth century. ...
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In more recent times, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences. For example, the partition number for every 17,303rd integer starting with 237 is divisible by 13. A. O. L. Atkin is a Professor Emeritus of mathematics at the University of Illinois at Chicago. ...
The University of Illinois at Chicago (UIC) is a public, state-supported research university. ...
References - Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (See chapter 5 for a modern pedagogical intro to Rademacher's formula).
- D. H. Lehmer, On the remainder and convergence of the series for the partition function Trans. Amer. Math. Soc. 46(1939) pp 362-373. (Provides the main formula (no derivatives), remainder, and older form for Ak(n).)
- Gupta, Gwyther, Miller, Roy. Soc. Math. Tables, vol 4, Tables of partitions, (1962) (Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for Ak(n) which is in Whiteman.)
- A. L. Whiteman, A sum connected with the series for the partition function, Pacific Journal of Math. 6:1 (1956) 159-176. (Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)
- Hans Rademacher, Collected Papers of Hans Rademacher, (1974) MIT Press; v II, p 100-107, 108-122, 460-475.
Hans Rademacher (1892 - 1969) was a German mathematician, known for work in mathematical analysis and number theory. ...
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