Integer partition

This article is about partitioning an integer. For grouping elements of a set, seePartition of a set.For the partition calculus of sets, seeInfinitary combinatorics.For the problem of partitioning a multiset of integers so that each part has the same sum, seePartition problem.

Young diagramsassociated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions.

Partitions ofnwith largest partk

Innumber theoryandcombinatorics,apartitionof a non-negativeintegern,also called aninteger partition,is a way of writingnas asumofpositive integers.Two sums that differ only in the order of theirsummandsare considered the same partition. (If order matters, the sum becomes acomposition.) For example,4can 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 + 3is the same partition as3 + 1,and the two distinct compositions1 + 2 + 1and1 + 1 + 2represent the same partition as2 + 1 + 1.

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

Partitions can be graphically visualized withYoung diagramsorFerrers diagrams.They occur in a number of branches ofmathematicsandphysics,including the study ofsymmetric polynomialsand of thesymmetric groupand ingroup representation theoryin 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 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$1^{m_{1}}2^{m_{2}}3^{m_{3}}\cdots$,wherem₁is the number of 1's,m₂is the number of 2's, etc. (Components withm_i= 0may 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$1^{5}$.

Diagrammatic representations of partitions

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.

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

Main article:Young diagram

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 functionsandgroup 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.As a type of shape made by adjacent squares joined together, Young diagrams are a special kind ofpolyomino.

Partition function

Using Euler's method to findp(40): A ruler with plus and minus signs (grey box) is slid downwards, the relevant parts added or subtracted. The positions of the signs are given by differences of alternating natural (blue) and odd (orange) numbers. Inthe SVG file,hover over the image to move the ruler.

Main article:Partition function (number theory)

Thepartition 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$n=0,1,2,\dots$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,... (sequenceA000041in theOEIS).

Thegenerating functionof$p$is

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

Noclosed-form expressionfor the partition function is known, but it has bothasymptotic expansionsthat accurately approximate it andrecurrence relationsby which it can be calculated exactly. It grows as anexponential functionof thesquare rootof its argument.,as follows:

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

In 1937,Hans Rademacherfound a way to represent the partition function$p(n)$by theconvergent series

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

$$A_{k}(n)=\sum _{0\leq m<k,\;(m,k)=1}e^{\pi i\left(s(m,k)-2nm/k\right)}.$$ and$s(m,k)$is theDedekind sum.

Themultiplicative inverseof its generating function is theEuler function;by Euler'spentagonal number theoremthis function is an alternating sum ofpentagonal numberpowers of its argument.

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

Srinivasa Ramanujandiscovered that the partition function has nontrivial patterns inmodular arithmetic,now known asRamanujan'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.

Restricted partitions

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 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 beconjugateof 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 are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to beself-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
Dist. odd		self-conjugate

Odd parts and distinct parts

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).This result was proved byLeonhard Eulerin 1748and 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,... (sequenceA000009in theOEIS).

Thegenerating functionforq(n) is given by

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

Thepentagonal number theoremgives a recurrence forq:

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_kis (−1)^mifk= 3m²−mfor some integermand is 0 otherwise.

Restricted part size or number of parts

Main article:Triangle of partition numbers

By taking conjugates, the numberp_k(n)of partitions ofninto exactlykparts is equal to the number of partitions ofnin 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) = 1andp_k(n) = 0ifn≤ 0 ork≤ 0andnandkare not both zero.

One recovers the functionp(n) by

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

One possible generating function for such partitions, takingkfixed andnvariable, is

$\sum _{n\geq 0}p_{k}(n)x^{n}=x^{k}\prod _{i=1}^{k}{\frac {1}{1-x^{i}}}.$

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

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

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

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

and the number of partitions ofnin which all parts are 1, 2 or 3 (or, equivalently, the number of partitions ofninto at most three parts) is the nearest integer to (n+ 3)²/ 12.

Partitions in a rectangle and Gaussian binomial coefficients

Main article:Gaussian binomial coefficient

One may also simultaneously limit the number and size of the parts. Letp(N,M;n)denote the number of partitions ofnwith at mostMparts, each of size at mostN.Equivalently, these are the partitions whose Young diagram fits inside anM×Nrectangle. 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 ofninto exactlyMparts of size at mostN,and subtracting 1 from each part of such a partition yields a partition ofn−Minto at mostMparts.

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

Rank and Durfee square

Main article:Durfee square

Therankof a partition is the largest numberksuch that the partition contains at leastkparts 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×rsquare 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.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$\lambda _{k}-k$for a partition ofkparts with largest part$\lambda _{k}$.This statistic (which is unrelated to the one described above) appears in the study ofRamanujan congruences.

Young's lattice

Main article:Young's lattice

There is a naturalpartial orderon 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 representationsofsymmetric groupsS_nfor 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.

Random partitions

There is a deep theory of random partitions chosen according to the uniform probability distribution on thesymmetric groupvia 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 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 theTracy–Widom distribution.Okounkovrelated these results to the combinatorics ofRiemann surfacesand representation theory.

See also

• Rank of a partition,a different notion ofrank
• 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 (seeGoldbach's conjecture)
• Kostant's partition function

References

• Abramowitz, Milton;Stegun, Irene(1964).Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.United States Department of Commerce, National Bureau of Standards.ISBN0-486-61272-4.
• Andrews, George E.(1976).The Theory of Partitions.Cambridge University Press.ISBN0-521-63766-X.
• Andrews, George E.; Eriksson, Kimmo (2004).Integer Partitions.Cambridge University Press.ISBN0-521-60090-1.
• 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.ISBN0-387-97127-0.Zbl0697.10023.(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.ISBN981-02-4900-4.(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-BrownandJ. H. Silverman.Foreword byAndrew Wiles.(6th ed.). Oxford:Oxford University Press.ISBN978-0-19-921986-5.MR2445243.Zbl1159.11001.
• 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.MR0000410.Zbl0022.20401.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.ISBN0-19-853530-9.Zbl0487.20007.(See section I.1)
• Nathanson, M.B. (2000).Elementary Methods in Number Theory.Graduate Texts in Mathematics. Vol. 195.Springer-Verlag.ISBN0-387-98912-9.Zbl0953.11002.
• 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.ISBN9780066210704.
• Stanley, Richard P.(1999).Enumerative Combinatorics.Vol. 1 and 2. Cambridge University Press.ISBN0-521-56069-1.
• 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.Zbl0071.04004.(Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)

External links

• "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 partitionswith reference tables to the On-Line Encyclopedia of Integer Sequences
• Integer partitionsentry in theFindStatdatabase
• Integer::Partition Perl modulefromCPAN
• Fast Algorithms For Generating Integer Partitions
• Generating All Partitions: A Comparison Of Two Encodings
• Grime, James (April 28, 2016)."Partitions - Numberphile"(video).Brady Haran.Archivedfrom the original on 2021-12-11.Retrieved5 May2016.