Integer partition#Odd parts and distinct parts

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 non-increasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the tuple {{math|(2, 2, 1)}} or in the even more compact form {{math|(2², 1)}} where the superscript indicates the number of repetitions of a part.

This multiplicity notation for a partition can be written alternatively as $1^\{m\_1\}2^\{m\_2\}3^\{m\_3\}\backslash cdots$, where {{math|m₁}} is the number of 1's, {{math|m₂}} is the number of 2's, etc. (Components with {{math|m_i {{=}} 0}} may be omitted.) For example, in this notation, the partitions of 5 are written $5^1,\; 1^1\; 4^1,\; 2^1\; 3^1,\; 1^2\; 3^1,\; 1^1\; 2^2,\; 1^3\; 2^1$, and $1^5$.

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.

The partition 6 + 4 + 3 + 1 of the number 14 can be represented by the following diagram:

File:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svg
File:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svg
File:GrayDot.svgFile:GrayDot.svgFile:GrayDot.svg
File:GrayDot.svg

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:

{{Main|Young diagram}}

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

:100px

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.{{sfn|Andrews|1976|p=199}} As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.{{citation

| last = Josuat-Vergès | first = Matthieu

| arxiv = 0801.4928

| doi = 10.1016/j.jcta.2010.03.006

| issue = 8

| journal = Journal of Combinatorial Theory

| mr = 2677686

| pages = 1218–1230

| series = Series A

| title = Bijections between pattern-avoiding fillings of Young diagrams

| volume = 117

| year = 2010| s2cid = 15392503

}}.

File:Euler_partition_function.svg

{{main|Partition function (number theory)}}

The partition function $p(n)$ counts the partitions of a non-negative integer $n$. For instance, $p(4)=5$ because the integer $4$ has the five partitions $1+1+1+1$, $1+1+2$, $1+3$, $2+2$, and $4$.

The values of this function for $n=0,1,2,\backslash dots$ 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, ... {{OEIS|id=A000041}}.

The generating function of $p$ is

:$\backslash sum\_\{n=0\}^\{\backslash infty\}p(n)q^n=\backslash prod\_\{j=1\}^\{\backslash infty\}\backslash sum\_\{i=0\}^\{\backslash infty\}q^\{ji\}=\backslash prod\_\{j=1\}^\{\backslash infty\}(1-q^j)^\{-1\}.$

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.,{{sfn|Andrews|1976|p=69}} as follows:

:$p(n)\; \backslash sim\; \backslash frac\; \{1\}\; \{4n\backslash sqrt3\}\; \backslash exp\backslash left(\{\backslash pi\; \backslash sqrt\; \{\backslash frac\{2n\}\{3\}\}\}\backslash right)$ as $n\; \backslash to\; \backslash infty$

In 1937, Hans Rademacher found a way to represent the partition function $p(n)$ by the convergent series

$$p(n)\; =\; \backslash frac\{1\}\{\backslash pi\; \backslash sqrt\{2\}\}\; \backslash sum\_\{k=1\}^\backslash infty\; A\_k(n)\backslash sqrt\{k\}\; \backslash cdot$$

\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)

where

e^{ \pi i \left( s(m, k) - 2 nm/k \right) }.

and $s(m,k)$ 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.

:$p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\backslash cdots$

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 $n$ ends in the digit 4 or 9, the number of partitions of $n$ will be divisible by 5.{{sfn|Hardy|Wright|2008|p=380}}

In both combinatorics and number theory, families of partitions subject to various restrictions are often studied.{{cite journal|last=Alder|first=Henry L.|title=Partition identities - from Euler to the present|journal=American Mathematical Monthly|volume=76|year=1969|issue=7|pages=733–746|url=http://www.maa.org/programs/maa-awards/writing-awards/partition-identities-from-euler-to-the-present|doi=10.2307/2317861|jstor=2317861}} This section surveys a few such restrictions.

{{anchor|Conjugate partitions}}

If we flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:

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.{{sfn|Hardy|Wright|2008|p=362}} 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.{{sfn|Hardy|Wright|2008|p=368}}

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:

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).{{sfn|Hardy|Wright|2008|p=365}}Notation follows {{harvnb|Abramowitz| Stegun|1964|p=825}} This result was proved by Leonhard Euler in 1748{{cite book|author-link=George Andrews (mathematician)|last=Andrews|first=George E.|title=Number Theory|publisher=W. B. Saunders Company|location=Philadelphia|date=1971|pages= 149–50}} 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, ... {{OEIS|id=A000009}}.

The generating function for q(n) is given by{{harvnb|Abramowitz|Stegun|1964|p=825}}, 24.2.2 eq. I(B)

:$\backslash sum\_\{n=0\}^\backslash infty\; q(n)x^n\; =\; \backslash prod\_\{k=1\}^\backslash infty\; (1+x^k)\; =\; \backslash prod\_\{k=1\}^\backslash infty\; \backslash frac\; \{1\}\{1-x^\{2k-1\}\}\; .$

The pentagonal number theorem gives a recurrence for q:{{harvnb|Abramowitz|Stegun|1964|p=826}}, 24.2.2 eq. II(A)

:q(k) = a_k + q(k − 1) + q(k − 2) − q(k − 5) − q(k − 7) + q(k − 12) + q(k − 15) − q(k − 22) − ...

where a_k is (−1)^m if k = 3m² − m for some integer m and is 0 otherwise.

{{main|Triangle of partition numbers}}

By taking conjugates, the number {{math|p_k(n)}} of partitions of {{math|n}} into exactly k parts is equal to the number of partitions of {{math|n}} in which the largest part has size {{math|k}}. The function {{math|p_k(n)}} satisfies the recurrence

: {{math|1=p_k(n) = p_k(n − k) + p_k−1(n − 1)}}

with initial values {{math|1=p₀(0) = 1}} and {{math|1=p_k(n) = 0}} if {{math|n ≤ 0 or k ≤ 0}} and {{math|n}} and {{math|k}} are not both zero.Richard Stanley, Enumerative Combinatorics, volume 1, second edition. Cambridge University Press, 2012. Chapter 1, section 1.7.

One recovers the function p(n) by

:$$

p(n) = \sum_{k = 0}^n p_k(n).

One possible generating function for such partitions, taking k fixed and n variable, is

: $\backslash sum\_\{n\; \backslash geq\; 0\}\; p\_k(n)\; x^n\; =\; x^k\backslash prod\_\{i\; =\; 1\}^k\; \backslash frac\{1\}\{1\; -\; x^i\}.$

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

:$\backslash prod\_\{t\; \backslash in\; T\}(1-x^t)^\{-1\}.$

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

:$\backslash left\; \backslash lfloor\; \backslash frac\{n\}\{2\}+1\; \backslash right\; \backslash rfloor\; ,$

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.{{cite book|last=Hardy|first=G.H.|title=Some Famous Problems of the Theory of Numbers|url=https://archive.org/details/in.ernet.dli.2015.84630|publisher=Clarendon Press|date=1920}}

{{Main|Gaussian binomial coefficient}}

One may also simultaneously limit the number and size of the parts. Let {{math|p(N, M; n)}} denote the number of partitions of {{mvar|n}} with at most {{mvar|M}} parts, each of size at most {{mvar|N}}. Equivalently, these are the partitions whose Young diagram fits inside an {{math|M × N}} rectangle. There is a recurrence relation

$$p(N,M;n)\; =\; p(N,M-1;n)\; +\; p(N-1,M;n-M)$$

obtained by observing that $p(N,M;n)\; -\; p(N,M-1;n)$ counts the partitions of {{mvar|n}} into exactly {{mvar|M}} parts of size at most {{mvar|N}}, and subtracting 1 from each part of such a partition yields a partition of {{math|n − M}} into at most {{mvar|M}} parts.{{sfn|Andrews|1976|pp=33–34}}

The Gaussian binomial coefficient is defined as:

$$\{k+\backslash ell\; \backslash choose\; \backslash ell\}\_q\; =\; \{k+\backslash ell\; \backslash choose\; k\}\_q\; =\; \backslash frac\{\backslash prod^\{k+\backslash ell\}\_\{j=1\}(1-q^j)\}\{\backslash prod^\{k\}\_\{j=1\}(1-q^j)\backslash prod^\{\backslash ell\}\_\{j=1\}(1-q^j)\}.$$

The Gaussian binomial coefficient is related to the generating function of {{math|p(N, M; n)}} by the equality

$$\backslash sum^\{MN\}\_\{n=0\}p(N,M;n)q^n\; =\; \{M+N\; \backslash choose\; M\}\_q.$$

{{main|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:

:

The Durfee square has applications within combinatorics in the proofs of various partition identities.see, e.g., {{harvnb|Stanley|1999|p=58}} 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 $\backslash lambda\_k\; -\; k$ for a partition of k parts with largest part $\backslash lambda\_k$. This statistic (which is unrelated to the one described above) appears in the study of Ramanujan congruences.

{{main|Young's lattice}}

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 S_n 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.

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 asymptotically 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.{{Cite book |last=Romik |first=Dan |title=The surprising mathematics of longest increasing subsequences |date=2015 |publisher=Cambridge University Press |isbn=978-1-107-42882-9 |series=Institute of Mathematical Statistics Textbooks |location=New York}} Okounkov related these results to the combinatorics of Riemann surfaces and representation theory.{{Cite journal |last=Okounkov |first=Andrei |date=2000 |title=Random matrices and random permutations |journal=International Mathematics Research Notices |volume=2000 |issue=20 |pages=1043 |doi=10.1155/S1073792800000532 |doi-access=|s2cid=14308256 }}{{Cite journal |last=Okounkov |first=A. |date=2001-04-01 |title=Infinite wedge and random partitions |url=https://doi.org/10.1007/PL00001398 |journal=Selecta Mathematica |language=en |volume=7 |issue=1 |pages=57–81 |doi=10.1007/PL00001398 |s2cid=119176413 |issn=1420-9020|arxiv=math/9907127 }}

{{Commons category|Integer partitions}}

{{div col|colwidth=30em}}

• 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

{{div col end}}

{{reflist}}

• {{cite book|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|first1=Milton|last1=Abramowitz|author1-link=Milton Abramowitz|first2=Irene|last2=Stegun|author2-link=Irene Stegun|publisher=United States Department of Commerce, National Bureau of Standards|isbn=0-486-61272-4|date=1964}}
• {{cite book|first=George E.|last=Andrews|author-link=George E. Andrews|title=The Theory of Partitions|date=1976|publisher=Cambridge University Press|isbn=0-521-63766-X}}
• {{cite book |first1=George E.|last1=Andrews|first2=Kimmo|last2=Eriksson |title=Integer Partitions |publisher=Cambridge University Press |year=2004 |isbn=0-521-60090-1}}
• {{cite book | last=Apostol | first=Tom M. | author-link=Tom M. Apostol | title=Modular functions and Dirichlet series in number theory | edition=2nd | series=Graduate Texts in Mathematics | volume=41 | location=New York etc. | publisher=Springer-Verlag | year=1990 | orig-year=1976 | isbn=0-387-97127-0 | zbl=0697.10023 | url-access=registration | url=https://archive.org/details/modularfunctions0000apos }} (See chapter 5 for a modern pedagogical intro to Rademacher's formula).
• {{cite book |first=Miklós|last=Bóna | author-link = Miklós Bóna |title=A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory |publisher=World Scientific Publishing |year=2002 |isbn=981-02-4900-4}} (an elementary introduction to the topic of integer partitions, including a discussion of Ferrers graphs)
• {{wikicite|reference={{Hardy and Wright|citation=cite book}}|ref={{harvid|Hardy|Wright|2008}}}}
• {{cite journal|first1=D. H.|last1=Lehmer|author1-link=D. H. Lehmer | title=On the remainder and convergence of the series for the partition function | journal=Trans. Amer. Math. Soc. | volume=46 | year=1939 | pages=362–373 | doi=10.1090/S0002-9947-1939-0000410-9 | mr=0000410 | zbl=0022.20401 | doi-access=free }} Provides the main formula (no derivatives), remainder, and older form for A_k(n).)
• {{cite book|first1=Hansraj|last1=Gupta|last2=Gwyther|first2=C.E.|last3=Miller|first3=J.C.P.|title=Royal Society of Math. Tables|volume=4, Tables of partitions|date=1962}} (Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for A_k(n), which is in Whiteman.)
• {{cite book | first=Ian G. | last=Macdonald | author-link=Ian G. Macdonald | title=Symmetric functions and Hall polynomials | series=Oxford Mathematical Monographs | publisher=Oxford University Press | year=1979 | isbn=0-19-853530-9 | zbl=0487.20007 }} (See section I.1)
• {{cite book | title=Elementary Methods in Number Theory | volume=195 | series=Graduate Texts in Mathematics | first=M.B. | last=Nathanson | publisher=Springer-Verlag | year=2000 | isbn=0-387-98912-9 | zbl=0953.11002 }}
• {{cite book|author-link=Hans Rademacher|first=Hans|last=Rademacher|title=Collected Papers of Hans Rademacher|date=1974|publisher=MIT Press|volume=v II|pages= 100–07, 108–22, 460–75}}
• {{cite book|author-link=Marcus du Sautoy|last=Sautoy|first=Marcus Du.|title=The Music of the Primes|url=https://archive.org/details/musicofprimessea00dusa|url-access=registration|location=New York|publisher=Perennial-HarperCollins|date=2003|isbn=9780066210704}}
• {{cite book|author-link=Richard P. Stanley|first=Richard P.|last=Stanley|url=http://www-math.mit.edu/~rstan/ec/|title=Enumerative Combinatorics|volume=1 and 2|publisher=Cambridge University Press|date=1999|isbn=0-521-56069-1}}
• {{cite journal | first=A. L. | last=Whiteman | url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044252 | title=A sum connected with the series for the partition function | journal=Pacific Journal of Mathematics | volume=6 | number=1 | year=1956 | pages=159–176 | doi=10.2140/pjm.1956.6.159 | zbl=0071.04004 | doi-access=free }} (Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)

• {{springer|title=Partition|id=p/p071740}}
• [http://www.se16.info/js/partitions.htm Partition and composition calculator]
• {{MathWorld | urlname=Partition | title=Partition }}
• Wilf, Herbert S. {{citation |url=http://www.math.upenn.edu/%7Ewilf/PIMS/PIMSLectures.pdf |title=Lectures on Integer Partitions |archive-url=https://web.archive.org/web/20210224220544if_/https://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf |archive-date=2021-02-24 |access-date=2021-02-28 |url-status=live }}
• [http://www.luschny.de/math/seq/CountingWithPartitions.html Counting with partitions] with reference tables to the On-Line Encyclopedia of Integer Sequences
• [http://www.findstat.org/IntegerPartitions Integer partitions] entry in the [http://www.findstat.org/ FindStat] database
• [http://metacpan.org/module/Integer::Partition Integer::Partition Perl module] from CPAN
• [https://web.archive.org/web/20090220004749/http://www.site.uottawa.ca/~ivan/F49-int-part.pdf Fast Algorithms For Generating Integer Partitions]
• [https://arxiv.org/abs/0909.2331 Generating All Partitions: A Comparison Of Two Encodings]
• {{cite web|last1=Grime|first1=James|title=Partitions - Numberphile|url=https://www.youtube.com/watch?v=NjCIq58rZ8I| archive-url=https://ghostarchive.org/varchive/youtube/20211211/NjCIq58rZ8I| archive-date=2021-12-11 | url-status=live|publisher=Brady Haran|access-date=5 May 2016|format=video|date=April 28, 2016}}{{cbignore}}