Partition (number theory)

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Young diagrams associated 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 of n with biggest addend k

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:

4
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 1 + 2 + 1 and 1 + 1 + 2 are the same partition as 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.

Examples[edit]

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

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.

Restricted partitions[edit]

A restricted partition is a partition in which the parts are constrained in some way.

For example, we could count partitions that contain only odd numbers. 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:

  • 8
  • 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[1] and is a special case of Glaisher's theorem.

Some similar results about restricted partitions can be obtained by the aid of a visual tool, a Ferrers graph (also called Ferrers diagram, since it is not a graph in the graph-theoretical sense, or sometimes Young diagram, alluding to the Young tableau).

Some results concerning restricted partitions are:

  • The number of partitions of n in which the greatest part is m is equal to the number of partitions of n into m parts.
  • The number of partitions of n in which each part is less than or equal to m is equal to the number of partitions of n into m or fewer parts.
  • The number of partitions of n in which all parts are equal is the number of divisors of n.
  • 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
\left \lfloor \frac {n}{2}+1 \right \rfloor \, .
  • The number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into 1, 2 or 3 parts) is the nearest integer to (n + 3)2 / 12.[2]

One class of restricted partition is specified by limiting the partitions to have at most M parts, each of size at most N. Let p(N,M;n) denote the number of such partitions of n. 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 n into exactly M parts of size at most N, and subtracting 1 from each part of such a partitions yields a partition of nM.[3]

Partition function[edit]

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, … (sequence A000041 in OEIS).

The value of p(n) has been computed for large values of n, for example p(100)=190,569,292 and p(1000) is 24,061,467,864,032,622,473,692,149,727,991 or approximately 2.4×1031.[4]

As of June 2013, the largest known prime number that counts a number of partitions is p(120052058), with 12198 decimal digits.[5]

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.[6] As noted above, q(n) is also the number of partitions of n into odd 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, … (sequence A000009 in OEIS).

Generating function[edit]

The generating function for p(n) is given by:[7]

\sum_{n=0}^\infty p(n)x^n = \prod_{k=1}^\infty \left(\frac {1}{1-x^k} \right).

Expanding each term 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.

The formulation of Euler's generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function.

The denominator of the product is Euler's function and can be written, by the pentagonal number theorem, as

(1-x)(1-x^2)(1-x^3) \dots = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} + \dots.

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 (-1)m. 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.

Another way of stating this is that the value of p(n) can be found from the formula[8]

 \begin{matrix}  {\rm GPN's} \\ 0 \\1\\2\\~\\~\\5\\~\\ 7\\ ~ \\ ~\\ \vdots \\ ~ \\ ~   \end{matrix}  
     ~~~    p(n) = \begin{vmatrix} ~~1 & -1~ & ~& ~ & ~ &~&~&~ \\
                                                         ~~1 & ~1 & -1~ & ~ \\
                                                         ~~0 & ~1 & ~1  & -1~ & ~ \\
                                                         ~~0 & ~0 & ~1 & ~1 &-1~ & ~ \\
                                                          -1 &~0 & ~0 & ~1 & ~1 &-1~ & ~  \\
                                                         ~~0 & -1~ & ~0 & ~0  & ~1 & ~1 & -1~ & ~ \\
                                                           -1 & ~0& -1~ & ~0 & ~0  & ~1 & ~1 & -1~ &~ \\ 
                                                         ~~0 & -1~ &~0& -1~ & ~0 & ~0  & ~1 & ~1 & -1~ &~ \\
                                                         ~~0 & ~0 & -1~ &~0& -1~ & ~0 & ~0  & ~1 & ~1 & ~ \\ 
                                                           ~~ \vdots & ~ & ~ & ~ & ~ & ~ &~ & ~ & ~ &  \ddots   \\ 
\end{vmatrix} _{ n \times n} .

I.e., p(n) is the determinant of the n×n truncation of the infinite-dimensional Toeplitz matrix shown above. The only non-zero diagonals of this matrix start on a row labeled by a generalized pentagonal number qm. (The superdiagonal is taken to start on row "0".) On these diagonals, the matrix element is (-1)m+1. This follows from a general formula for the quotients for power series.[9]

The generating function for q(n) (partitions into distinct parts) is given by:[10]

\sum_{n=0}^\infty q(n)x^n = \prod_{k=1}^\infty (1+x^k) = \prod_{k=1}^\infty \left(\frac {1}{1-x^{2k-1}} \right).

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:[11]

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.

The determinant formula for the quotient of power series can be applied to the expression ϕ(x2) / ϕ(x) to produce the expression

q(n) = \begin{vmatrix}   ~1& ~ & ~&~&~&~&~&~&~1~\\
                                                             -1& ~1& ~ & ~&~&~&~&~&~0~\\
                                                             -1& -1& ~1& ~ & ~&~&~&~&-1~\\
                                                             ~0& -1& -1& ~1 & ~ & ~&~&~&~0~\\
                                                             ~0 & ~0 & -1& -1&~1 & ~&~&~&-1~\\
                                                             ~1& ~0 & ~0& -1&   -1&~1&~&~& ~0~\\
                                                             ~0 &  ~1& ~0 & ~0& -1& -1&~1 &  ~ &~0~\\
                                                             ~1 &  ~0& ~1 & ~0&~0& -1& -1 &~&~0~\\   
                                                            ~ \vdots & ~&~&~&~&~& ~& \ddots &  ~\vdots~ \end{vmatrix}_{(n+1) \times (n+1)} ,

where the diagonals in the first n columns are constants equal to the coefficients in the power series for ϕ(x) and the last column has values ak given above.

Gaussian binomial coefficient[edit]

The Gaussian binomial coefficient is related to integer partitions. The Gaussian binomial coefficient is defined as:

{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)}.

The number of integer partitions that would fit into a k by l rectangle (when expressed as a Ferrers or Young diagram) is denoted by p(n, k, l). The Gaussian binomial coefficient is related to the generating function of p(n, k, l) by the following equality:

\sum^{k\ell}_{n=0}p(n,k,\ell)x^n = {k+\ell \choose \ell}_x.

Restricted partition generating functions[edit]

The generating function can be adapted to describe restricted partitions. For example, the generating function for integer partitions into distinct parts is:[12]

\prod^{\infty}_{n=1}(1+x^n)

and the generating function for partitions consisting of particular summands (specified by a set T of natural numbers) is:

\prod_{t \in T}(1-x^t)^{-1}.

This can be used to solve Change-making problems (where the set T specifies the available coins). Generating functions can be used to prove various identities involving integer partitions quite easily, for example the one mentioned in the Restricted partitions section. The generating function for partitions into odd summands is:[12]

\prod^{\infty}_{\begin{smallmatrix} n = 1 \\ n \mbox{ odd} \end{smallmatrix}}(1-x^n)^{-1} = \frac{1}{(1-x)(1-x^3)(1-x^5)...} = \frac{(1-x^2)(1-x^4)...}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)...}
 = \frac{(1-x)(1+x)(1-x^2)(1+x^2)...}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)...} = (1+x)(1+x^2)(1+x^3)...

which is the generating function for partitions into distinct summands.

The generating function for p(N,M;n) is a polynomial of degree NM: it is the Gaussian polynomial {M+N \choose M}_x where[13]

{m \choose r}_q
= \begin{cases}
\frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)} & r \le m \\
0 & r>m \end{cases}

Congruences[edit]

Srinivasa Ramanujan is credited with discovering that "congruences" in the number of partitions exist for arguments that are integers ending in 4 and 9.[14]

p(5k+4)\equiv 0 \pmod 5\,

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 an identity, also by Ramanujan,[15]

  \sum_{k=0}^{\infty} p(5k+4)x^k = 5~ \frac{ (x^5)^5_{\infty} } {(x)^6_{\infty}}

where the series (x)_{\infty} is defined as

(x)_{\infty} = \prod_{m=1}^{\infty}(1-x^m).

He also discovered congruences related to 7 and 11:[16]

\begin{align}
 p(7k + 5) &\equiv 0 \pmod 7\\
 p(11k + 6) &\equiv 0 \pmod {11}.
\end{align}

and for p=7 prove similar as above relation


  \sum_{k=0}^{\infty} p(7k+5)x^k =
   7~ \frac{ (x^7)^3_{\infty} } {(x)^4_{\infty}}
   +49 ~ \frac{ (x^7)^7_{\infty} } {(x)^8_{\infty}}

Since 5, 7, and 11 are consecutive primes, one might think that there would be such a congruence for the next prime 13, \scriptstyle p(13k \,+\, a) \;\equiv\; 0 \pmod{13} for some a. This is, however, false. It can also be shown that there is no congruence of the form \scriptstyle p(bk \,+\, a) \;\equiv\; 0 \pmod{b} for any prime b other than 5, 7, or 11.

In the 1960s, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences for small prime moduli. For example:

p(11^3 \cdot 13 \cdot k + 237)\equiv 0 \pmod {13}.

In 2000, Ken Ono of the University of Wisconsin–Madison proved that there are such congruences for every prime modulus. A few years later Ono, together with Scott Ahlgren of the University of Illinois, proved that there are partition congruences modulo every integer coprime to 6.[17]

Partition function formulas[edit]

Recurrence formula[edit]

Leonhard Euler's pentagonal number theorem implies the identity

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

where the numbers 1, 2, 5, 7, ... that appear on the right side of the equation are the generalized pentagonal numbers g_k  = \frac{k(3k -1)}{2} for nonzero integers k. More formally,

p(n)=\sum_k (-1)^{k-1}p\left(n- k(3k -1)/2\right)

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[edit]

Approximation formulas exist that are faster to calculate than the exact formula given above.

An asymptotic expression for p(n) is given by

p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right) \mbox { as } n\rightarrow \infty.

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:

p(n)=\frac{1}{2 \sqrt{2}} \sum_{k=1}^v \sqrt{k}\, A_k(n)\,
\frac{d}{dn} \exp \left({ \pi\sqrt{\frac23} 
    \frac{\sqrt{n-\frac{1}{24}}}{k} }
    \right)

where

A_k(n) = \sum_{0 \,\le\, m \,<\, k; \; (m,\, k) \,=\, 1}
e^{ \pi i \left[ s(m,\, k) \;-\; \frac{1}{k} 2 nm \right] }.

Here, the notation (mn) = 1 implies that the sum should occur only over the values of m that are relatively prime to n. The function s(mk) is a Dedekind sum.

The error after v terms is of the order of the next term, and v may be taken to be of the order of \sqrt n. 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.

In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is[18]

p(n)=\frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty \sqrt{k}\, A_k(n)\,
\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) .

The proof of Rademacher's formula involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way.

It may be shown that the k-th term of Rademacher's series is of the order

\exp\left(\pi\sqrt\frac23 \frac{\sqrt n}{k} \right) ,

so that the first term gives the Hardy–Ramanujan asymptotic approximation.

Paul Erdős published an elementary proof of the asymptotic formula for p(n) in 1942.[19][20]

Techniques for implementing the Hardy-Ramanujan-Rademacher formula efficiently on a computer are discussed in,[21] 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.[22]

Asymptotics of restricted partitions[edit]

The asymptotic expression for p(n) implies that

 \log p(n) \sim C \sqrt n \mbox { as } n\rightarrow \infty

where C = \pi\sqrt\frac23.[23]

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

 \log p_A(n) \sim C \sqrt{\alpha n}

and conversely if this asymptotic property holds for pA(n) then A has natural density α.[24] This result was stated, with a sketch of proof, by Erdős in 1942.[19][25]

If A is a finite set, this analysis does not apply (the density of a finite set is zero). If A has k elements then[26]

 p_A(n) = \left(\prod_{a \in A} a^{-1}\right) \cdot \frac{n^{k-1}}{(k-1)!} + O(n^{k-2}) .

Ferrers diagram[edit]

The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented by the following diagram; these diagrams are named in honor of Norman Macleod Ferrers:

****
***
***
**
*
*
6 + 4 + 3 + 1

The 14 circles are lined up in 4 columns, 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

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.[27] 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.[28]

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:

o*x
o*x
o*x
o*
o*
o*
o*
o
o
ooooo
o****
o*xx
o*x
o*
9 + 7 + 3 = 5 + 5 + 4 + 3 + 2
Dist. odd self-conjugate

Similar techniques can be employed to establish, for example, the following equalities:

  • The number of partitions of n into no more than k parts is the same as the number of partitions of n into parts no larger than k.
  • The number of partitions of n into no more than k parts is the same as the number of partitions of n + k into exactly k parts.

Young diagrams[edit]

An alternative visual representation of an integer partition is its Young diagram, named after the British mathematician Alfred Young. Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes. Thus, the Young diagram for the partition 5 + 4 + 1 is

Young diagram for 541 partition.svg

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.[29]

See also[edit]

Notes[edit]

  1. ^ Andrews, George E. Number Theory. W. B. Saunders Company, Philadelphia, 1971. Dover edition, page 149–150.
  2. ^ Hardy, G.H. Some Famous Problems of the Theory of Numbers. Clarendon Press, 1920.
  3. ^ Andrews (1976) pp.33-34
  4. ^ "Sloane's A070177 ", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ http://primes.utm.edu/top20/page.php?id=54
  6. ^ Notation follows Abramowitz and Stegun p. 825
  7. ^ Abramowitz and Stegun p. 825, 24.2.1 eq. I(B)
  8. ^ J. Malenfant, "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers"
  9. ^ The formula, due to Henri Faure, can be found in: Muir, Thomas (1920). The Theory of Determinants in the Historical Order of Development II. Macmillan and Co. p. 212. 
  10. ^ Abramowitz and Stegun p. 825, 24.2.2 eq. I(B)
  11. ^ Abramowitz and Stegun p. 826, 24.2.2 eq. II(A)
  12. ^ a b Hardy and Wright (2008) p.365
  13. ^ Andrews (1976) p.35
  14. ^ Hardy and Wright (2008) Theorem 359, p.380
  15. ^ Berndt and Ono, "Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary" [1]
  16. ^ Hardy and Wright (2008) Theorems 360,361, p.380
  17. ^ Ono, Ken; Ahlgren, Scott (2001). "Congruence properties for the partition function". Proceedings of the National Academy of Sciences 98 (23): 12,882–12,884. doi:10.1073/pnas.191488598. 
  18. ^ Andrews (1976) p.69
  19. ^ a b Erdős, Pál (1942). "On an elementary proof of some asymptotic formulas in the theory of partitions". Ann. Math. (2) 43: 437–450. Zbl 0061.07905. 
  20. ^ Nathanson (2000) p.456
  21. ^ F. Johansson, Efficient implementation of the Hardy-Ramanujan-Rademacher formula, LMS Journal of Computation and Mathematics 15 (2012), 341-359. [2]
  22. ^ Fredrik Johansson (March 2, 2014). "New partition function record: p(1020) computed". 
  23. ^ Andrews (1976) pp70,97
  24. ^ Nathanson (2000) pp.475–485
  25. ^ Nathanson (2000) p.495
  26. ^ Nathanson (2000) p.458–464
  27. ^ Hardy and Wright (2008) p.362
  28. ^ Hardy and Wright (2008) p.368
  29. ^ Andrews (1976) p.199

References[edit]

External links[edit]