Английская Википедия:Integer partition
Шаблон:Short description Шаблон:About
In number theory and combinatorics, a partition of a non-negative integer Шаблон:Mvar, also called an integer partition, is a way of writing Шаблон:Mvar 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, Шаблон:Math can be partitioned in five distinct ways:
The only partition of zero is the empty sum, having no parts.
The order-dependent composition Шаблон:Math is the same partition as Шаблон:Math, and the two distinct compositions Шаблон:Math and Шаблон:Math represent the same partition as Шаблон:Math.
An individual summand in a partition is called a part. The number of partitions of Шаблон:Mvar is given by the partition function Шаблон:Math. So Шаблон:Math. The notation Шаблон:Math means that Шаблон:Mvar is a partition of Шаблон:Mvar.
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 and of the symmetric group and in group representation theory in general.
Examples
The seven partitions of 5 are
- 5
- 4 + 1
- 3 + 2
- 3 + 1 + 1
- 2 + 2 + 1
- 2 + 1 + 1 + 1
- 1 + 1 + 1 + 1 + 1
Some authors treat a partition as a decreasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the tuple Шаблон:Math or in the even more compact form Шаблон:Math where the superscript indicates the number of repetitions of a part.
This multiplicity notation for a partition can be written alternatively as <math>1^{m_1}2^{m_2}3^{m_3}\cdots</math>, where Шаблон:Math is the number of 1's, Шаблон:Math is the number of 2's, etc. (Components with Шаблон:Math may be omitted.) For example, in this notation, the partitions of 5 are written <math>5^1, 1^1 4^1, 2^1 3^1, 1^2 3^1, 1^1 2^2, 1^3 2^1</math>, and <math>1^5</math>.
Diagrammatic 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 Alfred Young. Both have several possible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.
Ferrers diagram
The partition 6 + 4 + 3 + 1 of the 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 shown below:
| **** | *** * |
** ** |
** * * |
* * * * | ||||
| 4 | = | 3 + 1 | = | 2 + 2 | = | 2 + 1 + 1 | = | 1 + 1 + 1 + 1 |
Young diagram
Шаблон:Main An alternative visual representation of an integer partition is its Young diagram (often also called a Ferrers 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 does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of symmetric functions and group representation theory: 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.Шаблон:Sfn As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.[1]
Partition function
Шаблон:Main The partition function <math>p(n)</math> counts the partitions of a non-negative integer <math>n</math>. For instance, <math>p(4)=5</math> because the integer <math>4</math> has the five partitions <math>1+1+1+1</math>, <math>1+1+2</math>, <math>1+3</math>, <math>2+2</math>, and <math>4</math>. The values of this function for <math>n=0,1,2,\dots</math> are:
- 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, ... Шаблон:OEIS.
The generating function of <math>p</math> is
- <math>\sum_{n=0}^{\infty}p(n)q^n=\prod_{j=1}^{\infty}\sum_{i=0}^{\infty}q^{ji}=\prod_{j=1}^{\infty}(1-q^j)^{-1}.</math>
No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument.,Шаблон:Sfn as follows:
- <math>p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right)</math> as <math>n \to \infty</math>
In 1937, Hans Rademacher found a way to represent the partition function <math>p(n)</math> by the convergent series
<math display="block">p(n) = \frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty A_k(n)\sqrt{k} \cdot \frac{d}{dn} \left({
\frac {1} {\sqrt{n-\frac{1}{24}}}
\sinh \left[ {\frac{\pi}{k}
\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}}\,\,\,\right]
}\right)</math> where
<math display="block">A_k(n) = \sum_{0 \le m < k, \; (m, k) = 1} e^{ \pi i \left( s(m, k) - 2 nm/k \right) }.</math> and <math>s(m,k)</math> is the Dedekind sum.
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument.
- <math>p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots</math>
Srinivasa Ramanujan discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of <math>n</math> ends in the digit 4 or 9, the number of partitions of <math>n</math> will be divisible by 5.Шаблон:Sfn
Restricted partitions
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied.[2] This section surveys a few such restrictions.
Conjugate and self-conjugate partitions
Шаблон:Anchor If we 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.Шаблон:Sfn 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 are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to be self-conjugate.Шаблон:Sfn
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:
| ↔ | ooooo o**** o*xx o*x o* | |
| 9 + 7 + 3 | = | 5 + 5 + 4 + 3 + 2 |
| Dist. odd | self-conjugate |
Odd parts and distinct parts Шаблон:Anchor
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. Such a partition is called a partition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6:
- 8
- 7 + 1
- 6 + 2
- 5 + 3
- 5 + 2 + 1
- 4 + 3 + 1
This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n).Шаблон:Sfn[3] This result was proved by Leonhard Euler in 1748[4] and later was generalized as 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) (partitions into distinct parts). The first few values of q(n) are (starting with q(0)=1):
- 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... Шаблон:OEIS.
The generating function for q(n) is given by[5]
- <math>\sum_{n=0}^\infty q(n)x^n = \prod_{k=1}^\infty (1+x^k) = \prod_{k=1}^\infty \frac {1}{1-x^{2k-1}} .</math>
The pentagonal number theorem gives a recurrence for q:[6]
- 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
Шаблон:Main By taking conjugates, the number Шаблон:Math of partitions of Шаблон:Math into exactly k parts is equal to the number of partitions of Шаблон:Math in which the largest part has size Шаблон:Math. The function Шаблон:Math satisfies the recurrence
with initial values Шаблон:Math and Шаблон:Math if Шаблон:Math and Шаблон:Math and Шаблон:Math are not both zero.[7]
One recovers the function p(n) by
- <math>
p(n) = \sum_{k = 0}^n p_k(n). </math>
One possible generating function for such partitions, taking k fixed and n variable, is
- <math> \sum_{n \geq 0} p_k(n) x^n = x^k\prod_{i = 1}^k \frac{1}{1 - x^i}.</math>
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
- <math>\prod_{t \in T}(1-x^t)^{-1}.</math>
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
- <math>\left \lfloor \frac{n}{2}+1 \right \rfloor ,</math>
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.[8]
Partitions in a rectangle and Gaussian binomial coefficients
Шаблон:Main One may also simultaneously limit the number and size of the parts. Let Шаблон:Math denote the number of partitions of Шаблон:Mvar with at most Шаблон:Mvar parts, each of size at most Шаблон:Mvar. Equivalently, these are the partitions whose Young diagram fits inside an Шаблон:Math rectangle. There is a recurrence relation <math display=block>p(N,M;n) = p(N,M-1;n) + p(N-1,M;n-M)</math> obtained by observing that <math>p(N,M;n) - p(N,M-1;n)</math> counts the partitions of Шаблон:Mvar into exactly Шаблон:Mvar parts of size at most Шаблон:Mvar, and subtracting 1 from each part of such a partition yields a partition of Шаблон:Math into at most Шаблон:Mvar parts.Шаблон:Sfn
The Gaussian binomial coefficient is defined as: <math display=block>{k+\ell \choose \ell}_q = {k+\ell \choose k}_q = \frac{\prod^{k+\ell}_{j=1}(1-q^j)}{\prod^{k}_{j=1}(1-q^j)\prod^{\ell}_{j=1}(1-q^j)}.</math> The Gaussian binomial coefficient is related to the generating function of Шаблон:Math by the equality <math display=block>\sum^{MN}_{n=0}p(N,M;n)q^n = {M+N \choose M}_q.</math>
Rank and Durfee square
Шаблон:Main 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:
The Durfee square has applications within combinatorics in the proofs of various partition identities.[9] It also has some practical significance in the form of the h-index.
A different statistic is also sometimes called the rank of a partition (or Dyson rank), namely, the difference <math>\lambda_k - k</math> for a partition of k parts with largest part <math>\lambda_k</math>. This statistic (which is unrelated to the one described above) appears in the study of Ramanujan congruences.
Young's lattice
Шаблон:Main 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 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.
Random partitions
There is a deep theory of random partitions chosen according to the uniform probability distribution on the symmetric group via the Robinson–Schensted correspondence. In 1977, Logan and Shepp, as well as Vershik and Kerov, showed that the Young diagram of a typical large partition becomes asympototically close to the graph of a certain analytic function minimizing a certain functional. In 1988, Baik, Deift and Johansson extended these results to determine the distribution of the longest increasing subsequence of a random permutation in terms of the Tracy–Widom distribution.[10] Okounkov related these results to the combinatorics of Riemann surfaces and representation theory.[11][12]
See also
Шаблон:Commons category Шаблон:Div col
- Rank of a partition, a different notion of rank
- Crank of a partition
- Dominance order
- Factorization
- 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
- Multipartition
- Newton's identities
- Smallest-parts function
- A Goldbach partition is the partition of an even number into primes (see Goldbach's conjecture)
- Kostant's partition function
Notes
References
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book (See chapter 5 for a modern pedagogical intro to Rademacher's formula).
- Шаблон:Cite book (an elementary introduction to the topic of integer partitions, including a discussion of Ferrers graphs)
- Шаблон:Wikicite
- Шаблон:Cite journal Provides the main formula (no derivatives), remainder, and older form for Ak(n).)
- Шаблон:Cite book (Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for Ak(n), which is in Whiteman.)
- Шаблон:Cite book (See section I.1)
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite journal (Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)
External links
- Шаблон:Springer
- Partition and composition calculator
- Шаблон:MathWorld
- Wilf, Herbert S. Шаблон:Citation
- Counting with partitions with reference tables to the On-Line Encyclopedia of Integer Sequences
- Integer partitions entry in the FindStat database
- Integer::Partition Perl module from CPAN
- Fast Algorithms For Generating Integer Partitions
- Generating All Partitions: A Comparison Of Two Encodings
- Шаблон:Cite webШаблон:Cbignore
- ↑ Шаблон:Citation.
- ↑ Шаблон:Cite journal
- ↑ Notation follows Шаблон:Harvnb
- ↑ Шаблон:Cite book
- ↑ Шаблон:Harvnb, 24.2.2 eq. I(B)
- ↑ Шаблон:Harvnb, 24.2.2 eq. II(A)
- ↑ Richard Stanley, Enumerative Combinatorics, volume 1, second edition. Cambridge University Press, 2012. Chapter 1, section 1.7.
- ↑ Шаблон:Cite book
- ↑ see, e.g., Шаблон:Harvnb
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
