Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways:
- 3 + 1
- 2 + 2
- 2 + 1 + 1
- 1 + 1 + 1 + 1
The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1.
A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.
Partitions can be graphically visualized with Young diagrams or Ferrers 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.
- 1 Examples
- 2 Representations of partitions
- 3 Partition function
- 4 Restricted partitions
- 5 Rank and Durfee square
- 6 Young's lattice
- 7 Algorithm
- 8 See also
- 9 Notes
- 10 References
- 11 External links
The seven partitions of 5 are:
- 4 + 1
- 3 + 2
- 3 + 1 + 1
- 2 + 2 + 1
- 2 + 1 + 1 + 1
- 1 + 1 + 1 + 1 + 1
In some sources partitions are treated as the sequence of summands, rather than as an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the tuple (2, 2, 1) or in the even more compact form (22, 1) where the superscript indicates the number of repetitions of a term.
Representations of partitions
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after Norman Macleod Ferrers, and as Young diagrams, named after the British mathematician Alfred Young. Both have several possible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.
The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented by the following diagram:
The 14 circles are lined up in 4 rows, each having the size of a part of the partition. The diagrams for the 5 partitions of the number 4 are listed below:
|4||=||3 + 1||=||2 + 2||=||2 + 1 + 1||=||1 + 1 + 1 + 1|
An alternative visual representation of an integer partition is its Young diagram. Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for the partition 5 + 4 + 1 is
while the Ferrers diagram for the same partition is
While this seemingly trivial variation doesn't appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of symmetric functions and group representation theory: in particular, filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called Young tableaux, and these tableaux have combinatorial and representation-theoretic significance. As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.
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 ways of representing n as a sum of natural numbers (with order irrelevant). By convention p(0) = 1, p(n) = 0 for n negative.
The first few values of the partition function are (starting with p(0) = 1):
- 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, … (sequence A000041 in the OEIS).
The exact value of p(n) has been computed for large values of n, for example p(100) = 190,569,292, p(1000) is 24,061,467,864,032,622,473,692,149,727,991 or approximately 2.40615×1031, and p(10000) is 36,167,251,325,...,906,916,435,144 or approximately 3.61673×10106.
Expanding each factor on the right-hand side as a geometric series, we can rewrite it as
- (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.
where the exponents of x on the right hand side are the generalized pentagonal numbers; i.e., numbers of the form ½m(3m − 1), where m is an integer. The signs in the summation alternate as . This theorem can be used to derive a recurrence for the partition function:
- p(k) = p(k − 1) + p(k − 2) − p(k − 5) − p(k − 7) + p(k − 12) + p(k − 15) − p(k − 22) − ...
where p(0) is taken to equal 1, and p(k) is taken to be zero for negative k.
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. This is implied by the identity
He also discovered congruences related to 7 and 11:
and proved the identity
Since 5, 7, and 11 are consecutive primes, one might think that there would be such a congruence for the next prime 13, for some a. This is, however, false. In fact, it can be shown that there is no congruence of the form for any prime b other than 5, 7, or 11.
Partition function formulas
where the numbers 1, 2, 5, 7, ... that appear on the right side of the equation are the generalized pentagonal numbers for nonzero integers k. More formally,
where the summation is over all nonzero integers k (positive and negative) and p(m) is taken to be 0 if m < 0.
Approximation formulas exist that are faster to calculate than the exact formula given above.
An asymptotic expression for p(n) is given by
- as .
This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. Considering p(1000), the asymptotic formula gives about 2.4402 × 1031, reasonably close to the exact answer given above (1.415% larger than the true value).
Hardy and Ramanujan obtained an asymptotic expansion with this approximation as the first term:
The error after v terms is of the order of the next term, and v may be taken to be of the order of . As an example, Hardy and Ramanujan showed that p(200) is the nearest integer to the sum of the first v=5 terms of the series.
It may be shown that the k-th term of Rademacher's series is of the order
so that the first term gives the Hardy–Ramanujan asymptotic approximation.
Techniques for implementing the Hardy–Ramanujan–Rademacher formula efficiently on a computer are discussed in Johansson, where it is shown that p(n) can be computed in softly optimal time O(n1/2+ε). The largest value of the partition function computed exactly is p(1020), which has slightly more than 11 billion digits.
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied. This section surveys a few such restrictions.
Conjugate and self-conjugate partitions
If we now flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:
|6 + 4 + 3 + 1||=||4 + 3 + 3 + 2 + 1 + 1|
By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to be conjugate of one another. In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest is the partition 2 + 2, which has itself as conjugate. Such a partition is said to be self-conjugate.
Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugate diagram:
One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:
|9 + 7 + 3||=||5 + 5 + 4 + 3 + 2|
Odd parts and distinct parts
Among the 22 partitions of the number 8, there are 6 that contain only odd parts:
- 7 + 1
- 5 + 3
- 5 + 1 + 1 + 1
- 3 + 3 + 1 + 1
- 3 + 1 + 1 + 1 + 1 + 1
- 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Alternatively, we could count partitions in which no number occurs more than once. If we count the partitions of 8 with distinct parts, we also obtain 6:
- 7 + 1
- 6 + 2
- 5 + 3
- 5 + 2 + 1
- 4 + 3 + 1
For all positive numbers the number of partitions with odd parts equals the number of partitions with distinct parts. This result was proved by Leonhard Euler in 1748 and is a special case of Glaisher's theorem.
For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is q(n), the number of partitions of n into distinct parts. The first few values of q(n) are (starting with q(0)=1):
The second product can be written ϕ(x2) / ϕ(x) where ϕ is Euler's function; the pentagonal number theorem can be applied to this as well giving a recurrence for q:
- q(k) = ak + q(k − 1) + q(k − 2) − q(k − 5) − q(k − 7) + q(k − 12) + q(k − 15) − q(k − 22) − ...
where ak is (−1)m if k = 3m2 − m for some integer m and is 0 otherwise.
Restricted part size or number of parts
Using the same conjugation trick as above, one may show that the number pk(n) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k. The function pk(n) satisfies the recurrence
- pk(n) = pk(n − k) + pk − 1(n − 1)
with initial values p0(0) = 1 and pk(n) = 0 if n ≤ 0 or k ≤ 0.
One possible generating function for such partitions, taking k fixed and n variable, is
More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function
This can be used to solve change-making problems (where the set T specifies the available coins). As two particular cases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitions of n into 1 or 2 parts) is
and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to (n + 3)2 / 12.
The asymptotic expression for p(n) implies that
If A is a set of natural numbers, we let pA(n) denote the number of partitions of n into elements of A. If A possesses positive natural density α then
If A is a finite set, this analysis does not apply (the density of a finite set is zero). If A has k elements whose greatest common divisor is 1, then
Partitions in a rectangle and Gaussian binomial coefficients
One may also simultaneously limit the number and size of the parts. Let p(N, M; n) denote the number of partitions of n with at most M parts, each of size at most N. Equivalently, these are the partitions whose Young diagram fits inside an M × N rectangle. There is a recurrence relation
obtained by observing that counts the partitions of n into exactly M parts of size at most N, and subtracting 1 from each part of such a partitions yields a partition of n − M into at most M parts.
The Gaussian binomial coefficient is defined as:
The Gaussian binomial coefficient is related to the generating function of p(N, M; n) by the equality
Rank and Durfee square
The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank r, the r × r square of entries in the upper-left is known as the Durfee square:
There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set is known as Young's lattice. The lattice was originally defined in the context of representation theory, where it is used to describe of the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a differential poset.
The partition function is inherently recursive in nature since the results of smaller numbers appear as components in the result of a larger number.
Let p(n,m) be the number of partitions of n using only positive integers that are less than or equal to m.
It may be seen that p(n) = p(n,n), and also p(n,m) = p(n,n) = p(n) for m > n.
This recursive summation formula can be implemented with table lookups or memoization to remove duplicated effort.
|Wikimedia Commons has media related to Integer partitions.|
- Rank of a partition, a different notion of rank
- Crank of a partition
- Dominance order
- Integer factorization
- Partition of a set
- Stars and bars (combinatorics)
- Plane partition
- Polite number, defined by partitions into consecutive integers
- Multiplicative partition
- Twelvefold way
- Ewens's sampling formula
- Faà di Bruno's formula
- Newton's identities
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- Smallest-parts function
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