Innumber theory andcombinatorics, apartition of a non-negativeintegern, also called aninteger partition, is a way of writingn as asum ofpositive integers. Two sums that differ only in the order of theirsummands are considered the same partition. (If order matters, the sum becomes acomposition.) For example,4 can be partitioned in five distinct ways:

4

3 + 1

2 + 2

2 + 1 + 1

1 + 1 + 1 + 1

The only partition of zero is the empty sum, having no parts.

The order-dependent composition1 + 3 is the same partition as3 + 1, and the two distinct compositions1 + 2 + 1 and1 + 1 + 2 represent the same partition as2 + 1 + 1.

An individual summand in a partition is called apart. The number of partitions ofn is given by thepartition functionp(n). Sop(4) = 5. The notationλ ⊢n means thatλ is a partition ofn.

Partitions can be graphically visualized withYoung diagrams orFerrers diagrams. They occur in a number of branches ofmathematics andphysics, including the study ofsymmetric polynomials and of thesymmetric group and ingroup representation theory in general.

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 thetuple(2, 2, 1) or in the even more compact form(2², 1) where the superscript indicates the number of repetitions of a part.

This multiplicity notation for a partition can be written alternatively as${\displaystyle 1^{m_1}{2}^{m_2}{3}^{m_3}\cdots }$, wherem₁ is the number of 1's,m₂ is the number of 2's, etc. (Components withm_i = 0 may be omitted.) For example, in this notation, the partitions of 5 are written${\displaystyle 5^1,1^1{4}^1,2^1{3}^1,1^2{3}^1,1^1{2}^2,1^3{2}^1}$, and${\displaystyle 1^5}$.

There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named afterNorman Macleod Ferrers, and as Young diagrams, named afterAlfred Young. Both have several possible conventions; here, we useEnglish 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:

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

An alternative visual representation of an integer partition is itsYoung 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 ofsymmetric functions andgroup representation theory: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects calledYoung tableaux, and these tableaux have combinatorial and representation-theoretic significance.^[1] As a type of shape made by adjacent squares joined together, Young diagrams are a special kind ofpolyomino.^[2]

Thepartition function${\displaystyle p(n)}$ counts the partitions of a non-negative integer${\displaystyle n}$. For instance,${\displaystyle p(4)=5}$ because the integer${\displaystyle 4}$ has the five partitions${\displaystyle 1+1+1+1}$,${\displaystyle 1+1+2}$,${\displaystyle 1+3}$,${\displaystyle 2+2}$, and${\displaystyle 4}$.The values of this function for${\displaystyle n=0,1,2,\dots }$ 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, ... (sequenceA000041 in theOEIS).

Thegenerating function of${\displaystyle p}$ is

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

Noclosed-form expression for the partition function is known, but it has bothasymptotic expansions that accurately approximate it andrecurrence relations by which it can be calculated exactly. It grows as anexponential function of thesquare root of its argument.,^[3] as follows:

${\displaystyle p(n)\sim \frac{1}{4n\sqrt{3}}\exp \left(\pi \sqrt{\frac{2n}{3}}\right)}$ as${\displaystyle n\to \mathrm{\infty }}$

In 1937,Hans Rademacher found a way to represent the partition function${\displaystyle p(n)}$ by theconvergent series

$${\displaystyle p(n)=\frac{1}{\pi \sqrt{2}}\sum _{k=1}^{\mathrm{\infty }}{A}_k(n)\sqrt{k}\cdot \frac{d}{dn}\left(\frac{1}{\sqrt{n-\frac{1}{24}}}\sinh \left[\frac{\pi }{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right]\right)}$$where

and${\displaystyle s(m,k)}$ is theDedekind sum.

Themultiplicative inverse of its generating function is theEuler function; by Euler'spentagonal number theorem this function is an alternating sum ofpentagonal number powers of its argument.

${\displaystyle p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots }$

Srinivasa Ramanujan discovered that the partition function has nontrivial patterns inmodular arithmetic, now known asRamanujan's congruences. For instance, whenever the decimal representation of${\displaystyle n}$ ends in the digit 4 or 9, the number of partitions of${\displaystyle n}$ will be divisible by 5.^[4]

In both combinatorics and number theory, families of partitions subject to various restrictions are often studied.^[5] This section surveys a few such restrictions.

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 beconjugate of one another.^[6] 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 beself-conjugate.^[7]

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
Dist. odd		self-conjugate

Among the 22 partitions of the number 8, there are 6 that contain onlyodd 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 apartition 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 byq(n).^[8][9] This result was proved byLeonhard Euler in 1748^[10] and later was generalized asGlaisher'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 isq(n) (partitions into distinct parts). The first few values ofq(n) are (starting withq(0)=1):

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... (sequenceA000009 in theOEIS).

Thegenerating function forq(n) is given by^[11]

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

Thepentagonal number theorem gives a recurrence forq:^[12]

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) − ...

wherea_k is (−1)^m ifk = 3m² −m for some integerm and is 0 otherwise.

By taking conjugates, the numberp_k(n) of partitions ofn into exactlyk parts is equal to the number of partitions ofn in which the largest part has sizek. The functionp_k(n) satisfies the recurrence

p_k(n) =p_k(n −k) +p_k−1(n − 1)

with initial valuesp₀(0) = 1 andp_k(n) = 0 ifn ≤ 0 ork ≤ 0 andn andk are not both zero.^[13]

One recovers the functionp(n) by

${\displaystyle p(n)=\sum _{k=0}^n{p}_k(n).}$

One possible generating function for such partitions, takingk fixed andn variable, is

${\displaystyle \sum _{n\ge 0}{p}_k(n)x^n=x^k\prod _{i=1}^k\frac{1}{1-x^i}.}$

More generally, ifT is a set of positive integers then the number of partitions ofn, all of whose parts belong toT, hasgenerating function

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

This can be used to solvechange-making problems (where the setT specifies the available coins). As two particular cases, one has that the number of partitions ofn in which all parts are 1 or 2 (or, equivalently, the number of partitions ofn into 1 or 2 parts) is

${\displaystyle \lfloor \frac{n}{2}+1\rfloor ,}$

and the number of partitions ofn in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions ofn into at most three parts) is the nearest integer to (n + 3)² / 12.^[14]

One may also simultaneously limit the number and size of the parts. Letp(N,M;n) denote the number of partitions ofn with at mostM parts, each of size at mostN. Equivalently, these are the partitions whose Young diagram fits inside anM ×N rectangle. There is a recurrence relation$${\displaystyle p(N,M;n)=p(N,M-1;n)+p(N-1,M;n-M)}$$obtained by observing that${\displaystyle p(N,M;n)-p(N,M-1;n)}$ counts the partitions ofn into exactlyM parts of size at mostN, and subtracting 1 from each part of such a partition yields a partition ofn −M into at mostM parts.^[15]

The Gaussian binomial coefficient is defined as:$${\displaystyle {(\genfrac{}{}{0ex}{}{k+\ell }{\ell })}_q={(\genfrac{}{}{0ex}{}{k+\ell }{k})}_q=\frac{\prod _{j=1}^{k+\ell }(1-q^j)}{\prod _{j=1}^k(1-q^j)\prod _{j=1}^{\ell }(1-q^j)}.}$$The Gaussian binomial coefficient is related to thegenerating function ofp(N,M;n) by the equality$${\displaystyle \sum _{n=0}^{MN}p(N,M;n)q^n={(\genfrac{}{}{0ex}{}{M+N}{M})}_q.}$$

Therank of a partition is the largest numberk such that the partition contains at leastk parts of size at leastk. 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 rankr, ther ×r square of entries in the upper-left is known as theDurfee square:

The Durfee square has applications within combinatorics in the proofs of various partition identities.^[16] It also has some practical significance in the form of theh-index.

A different statistic is also sometimes called therank of a partition (or Dyson rank), namely, the difference${\displaystyle {\lambda }_k-k}$ for a partition ofk parts with largest part${\displaystyle {\lambda }_k}$. This statistic (which is unrelated to the one described above) appears in the study ofRamanujan congruences.

There is a naturalpartial order on partitions given by inclusion of Young diagrams. This partially ordered set is known asYoung's lattice. The lattice was originally defined in the context ofrepresentation theory, where it is used to describe theirreducible representations ofsymmetric groupsS_n for alln, 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 adifferential poset.

There is a deep theory of random partitions chosen according to the uniform probability distribution on thesymmetric group via theRobinson–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 theTracy–Widom distribution.^[17]Okounkov related these results to the combinatorics ofRiemann surfaces and representation theory.^[18][19]

Wikimedia Commons has media related toInteger partitions.

• Abramowitz, Milton;Stegun, Irene (1964).Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. United States Department of Commerce, National Bureau of Standards.ISBN 0-486-61272-4.
• Andrews, George E. (1976).The Theory of Partitions. Cambridge University Press.ISBN 0-521-63766-X.
• Andrews, George E.; Eriksson, Kimmo (2004).Integer Partitions. Cambridge University Press.ISBN 0-521-60090-1.
• Apostol, Tom M. (1990) [1976].Modular functions and Dirichlet series in number theory.Graduate Texts in Mathematics. Vol. 41 (2nd ed.). New York etc.:Springer-Verlag.ISBN 0-387-97127-0.Zbl 0697.10023.(See chapter 5 for a modern pedagogical intro to Rademacher's formula).
• Bóna, Miklós (2002).A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory. World Scientific Publishing.ISBN 981-02-4900-4. (an elementary introduction to the topic of integer partitions, including a discussion of Ferrers graphs)
• Hardy, G. H.;Wright, E. M. (2008) [1938].An Introduction to the Theory of Numbers. Revised byD. R. Heath-Brown andJ. H. Silverman. Foreword byAndrew Wiles. (6th ed.). Oxford:Oxford University Press.ISBN 978-0-19-921986-5.MR 2445243.Zbl 1159.11001.
• Lehmer, D. H. (1939)."On the remainder and convergence of the series for the partition function".Trans. Amer. Math. Soc.46:362–373.doi:10.1090/S0002-9947-1939-0000410-9.MR 0000410.Zbl 0022.20401. Provides the main formula (no derivatives), remainder, and older form forA_k(n).)
• Gupta, Hansraj; Gwyther, C.E.; Miller, J.C.P. (1962).Royal Society of Math. Tables. Vol. 4, Tables of partitions.(Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula forA_k(n),which is in Whiteman.)
• Macdonald, Ian G. (1979).Symmetric functions and Hall polynomials. Oxford Mathematical Monographs.Oxford University Press.ISBN 0-19-853530-9.Zbl 0487.20007. (See section I.1)
• Nathanson, M.B. (2000).Elementary Methods in Number Theory. Graduate Texts in Mathematics. Vol. 195.Springer-Verlag.ISBN 0-387-98912-9.Zbl 0953.11002.
• Rademacher, Hans (1974).Collected Papers of Hans Rademacher. Vol. v II. MIT Press. pp. 100–07,108–22,460–75.
• Sautoy, Marcus Du. (2003).The Music of the Primes. New York: Perennial-HarperCollins.ISBN 9780066210704.
• Stanley, Richard P. (1999).Enumerative Combinatorics. Vol. 1 and 2. Cambridge University Press.ISBN 0-521-56069-1.
• Whiteman, A. L. (1956)."A sum connected with the series for the partition function".Pacific Journal of Mathematics.6 (1):159–176.doi:10.2140/pjm.1956.6.159.Zbl 0071.04004.(Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)

• "Partition",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Partition and composition calculator
• Weisstein, Eric W."Partition".MathWorld.
• Wilf, Herbert S.Lectures on Integer Partitions(PDF),archived(PDF) from the original on 2021-02-24, retrieved2021-02-28
• Counting with partitions with reference tables to the On-Line Encyclopedia of Integer Sequences
• Integer partitions entry in theFindStat database
• Integer::Partition Perl module fromCPAN
• Fast Algorithms For Generating Integer Partitions
• Generating All Partitions: A Comparison Of Two Encodings
• Grime, James (April 28, 2016)."Partitions - Numberphile"(video).Brady Haran.Archived from the original on 2021-12-11. Retrieved5 May 2016.

• Integer partitions

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