Crank of a partition

Last updated
Freeman Dyson in 2005 Freeman Dyson.jpg
Freeman Dyson in 2005

In number theory, the crank of an integer partition is a certain number associated with the partition. It was first introduced without a definition by Freeman Dyson, who hypothesised its existence in a 1944 paper. [1] Dyson gave a list of properties this yet-to-be-defined quantity should have. In 1988, George E. Andrews and Frank Garvan discovered a definition for the crank satisfying the properties hypothesized for it by Dyson. [2]

Contents

Dyson's crank

Let n be a non-negative integer and let p(n) denote the number of partitions of n (p(0) is defined to be 1). Srinivasa Ramanujan in a paper [3] published in 1918 stated and proved the following congruences for the partition function p(n), since known as Ramanujan congruences.

These congruences imply that partitions of numbers of the form 5n + 4 (respectively, of the forms 7n + 5 and 11n + 6 ) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruences were based on the ideas of generating functions and they did not specify a method for the division of the partitions into subclasses of equal size.

In his Eureka paper Dyson proposed the concept of the rank of a partition. The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = −1. Denoting by N(m, q, n), the number of partitions of n whose ranks are congruent to m modulo q, Dyson considered N(m, 5, 5 n + 4) and N(m, 7, 7n + 5) for various values of n and m. Based on empirical evidences Dyson formulated the following conjectures known as rank conjectures.

For all non-negative integers n we have:

Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form 5n + 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each other modulo 5. The same idea can be applied to divide the partitions of integers of the form 7n + 5 into seven equally numerous classes. But the idea fails to divide partitions of integers of the form 11n + 6 into 11 classes of the same size, as the following table shows.

Partitions of the integer 6 ( 11n + 6 with n = 0 ) divided into classes based on ranks
rank ≡ 0
(mod 11)
rank ≡ 1
(mod 11)
rank ≡ 2
(mod 11)
rank ≡ 3
(mod 11)
rank ≡ 4
(mod 11)
rank ≡ 5
(mod 11)
rank ≡ 6
(mod 11)
rank ≡ 7
(mod 11)
rank ≡ 8
(mod 11)
rank ≡ 9
(mod 11)
rank ≡ 10
(mod 11)
{3,2,1}{4,1,1}{4,2}{5,1}{6}{1,1,1,1,1,1}{2,1,1,1,1}{2,2,1,1}{2,2,2}
{3,3}{3,1,1,1}

Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,

I hold in fact :

Whether these guesses are warranted by evidence, I leave it to the reader to decide. Whatever the final verdict of posterity may be, I believe the "crank" is unique among arithmetical functions in having been named before it was discovered. May it be preserved from the ignominious fate of the planet Vulcan.

Definition of crank

In a paper [2] published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows:

For a partition λ, let (λ) denote the largest part of λ, ω(λ) denote the number of 1's in λ, and μ(λ) denote the number of parts of λ larger than ω(λ). The crank c(λ) is given by

The cranks of the partitions of the integers 4, 5, 6 are computed in the following tables.

Cranks of the partitions of 4
Partition
λ
Largest part
(λ)
Number of 1's
ω(λ)
Number of parts
larger than ω(λ)
μ(λ)
Crank
c(λ)
{4}4014
{3,1}3110
{2,2}2022
{2,1,1}220−2
{1,1,1,1}140−4
Cranks of the partitions of 5
Partition
λ
Largest part
(λ)
Number of 1's
ω(λ)
Number of parts
larger than ω(λ)
μ(λ)
Crank
c(λ)
{5}5015
{4,1}4110
{3,2}3023
{3,1,1}321−1
{2,2,1}2121
{2,1,1,1}230−3
{1,1,1,1,1}150−5
Cranks of the partitions of 6
Partition
λ
Largest part
(λ)
Number of 1's
ω(λ)
Number of parts
larger than ω(λ)
μ(λ)
Crank
c(λ)
{6}6016
{5,1}5110
{4,2}4024
{4,1,1}421−1
{3,3}3023
{3,2,1}3121
{3,1,1,1}330−3
{2,2,2}2032
{2,2,1,1}220−2
{2,1,1,1,1}240−4
{1,1,1,1,1,1}160−6


Notations

For all integers n ≥ 0 and all integers m, the number of partitions of n with crank equal to m is denoted by M(m,n) except for n = 1 where M(−1,1) = −M(0,1) = M(1,1) = 1 as given by the following generating function. The number of partitions of n with crank equal to m modulo q is denoted by M(m,q,n).

The generating function for M(m,n) is given below:

Basic result

Andrews and Garvan proved the following result [2] which shows that the crank as defined above does meet the conditions given by Dyson.

The concepts of rank and crank can both be used to classify partitions of certain integers into subclasses of equal size. However the two concepts produce different subclasses of partitions. This is illustrated in the following two tables.

Classification of the partitions of the integer 9 based on cranks
Partitions with
crank ≡ 0
(mod 5)
Partitions with
crank ≡ 1
(mod 5)
Partitions with
crank ≡ 2
(mod 5)
Partitions with
crank ≡ 3
(mod 5)
Partitions with
crank 4
(mod 5)
{ 8, 1 }{ 6, 3 }{ 7, 2 }{ 6, 1, 1, 1 }{ 9 }
{ 5, 4 }{ 6, 2, 1 }{ 5, 1, 1, 1, 1 }{ 4, 2, 1, 1, 1 }{ 7, 1, 1 }
{ 5, 2, 2 }{ 5, 3, 1 }{ 4, 2, 2, 1 }{ 3, 3, 3 }{ 5, 2, 1, 1 }
{ 4, 3, 1, 1 }{ 4, 4, 1 }{ 3, 3, 2, 1 }{ 3, 2, 2, 2 }{ 4, 3, 2 }
{ 4, 1, 1, 1, 1, 1 }{ 3, 2, 1, 1, 1, 1 }{ 3, 3, 1, 1, 1 }{ 2, 2, 2, 2, 1 }{ 3, 2, 2, 1, 1 }
{ 2, 2, 1, 1, 1, 1, 1 }{ 1, 1, 1, 1, 1, 1, 1, 1, 1 }{ 2, 2, 2, 1, 1, 1 }{ 2, 1, 1, 1, 1, 1, 1, 1}{ 3, 1, 1, 1, 1, 1, 1 }
Classification of the partitions of the integer 9 based on ranks
Partitions with
rank 0
(mod 5)
Partitions with
rank 1
(mod 5)
Partitions with
rank 2
(mod 5)
Partitions with
rank 3
(mod 5)
Partitions with
rank 4
(mod 5)
{ 7, 2 }{ 8, 1 }{ 6, 1, 1, 1 }{ 9 }{ 7, 1, 1 }
{ 5, 1, 1, 1, 1 }{ 5, 2, 1, 1 }{ 5, 3, 1}{ 6, 2, 1 }{ 6, 3 }
{ 4, 3, 1, 1 }{ 4, 4, 1 }{ 5, 2, 2 }{ 5, 4 }{ 4, 2, 1, 1, 1 }
{ 4, 2, 2, 1 }{ 4, 3, 2 }{ 3, 2, 1, 1, 1, 1 }{ 3, 3, 1, 1, 1 }{ 3, 3, 2, 1 }
{ 3, 3, 3 }{ 3, 1, 1, 1, 1, 1, 1 }{ 2, 2, 2, 2, 1 }{ 4, 1, 1, 1, 1, 1 }{ 3, 2, 2, 2 }
{ 2, 2, 1, 1, 1, 1, 1 }{ 2, 2, 2, 1, 1, 1 }{ 1, 1, 1, 1, 1, 1, 1, 1, 1 }{ 3, 2, 2, 1, 1}{ 2, 1, 1, 1, 1, 1, 1, 1 }

Ramanujan and cranks

Recent work by Bruce C. Berndt and his coauthors argued that Ramanujan knew about the crank, although not in the form that Andrews and Garvan have defined. In a systematic study of the Lost Notebook of Ramanujan, Berndt and his coauthors have given substantial evidence that Ramanujan knew about the dissections of the crank generating function. [4] [5]

Related Research Articles

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.

<span class="mw-page-title-main">Modular arithmetic</span> Computation modulo a fixed integer

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

<span class="mw-page-title-main">Quadratic reciprocity</span> Gives conditions for the solvability of quadratic equations modulo prime numbers

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

<span class="mw-page-title-main">Gaussian integer</span> Complex number whose real and imaginary parts are both integers

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or

<span class="mw-page-title-main">Integer partition</span> Decomposition of an integer as a sum of positive integers

In number theory and combinatorics, a partition of a non-negative 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. For example, 4 can be partitioned in five distinct ways:

In mathematics, a modular form is a (complex) analytic function on the upper half-plane, , that satisfies:

<span class="mw-page-title-main">Partition function (number theory)</span> The number of partitions of an integer

In number theory, the partition functionp(n) represents the number of possible 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.

In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

<span class="mw-page-title-main">Modular group</span> Orientation-preserving mapping class group of the torus

In mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

<span class="mw-page-title-main">Carmichael function</span> Function in mathematical number theory

In number theory, a branch of mathematics, the Carmichael functionλ(n) of a positive integer n is the smallest member of the set of positive integers m having the property that

Multiplicative group of integers modulo <i>n</i> Group of units of the ring of integers modulo n

In modular arithmetic, the integers coprime to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n.

<span class="mw-page-title-main">Ramanujan tau function</span>

The Ramanujan tau function, studied by Ramanujan, is the function defined by the following identity:

The Tonelli–Shanks algorithm is used in modular arithmetic to solve for r in a congruence of the form r2n, where p is a prime: that is, to find a square root of n modulo p.

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3p is solvable if and only if x3q is solvable.

In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) of type discovered by mathematician Srinivasa Ramanujan:

The spt function is a function in number theory that counts the sum of the number of smallest parts in each integer partition of a positive integer. It is related to the partition function.

In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation . If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. See modular arithmetic for notation and terminology.

<span class="mw-page-title-main">Rank of a partition</span>

In mathematics, particularly in the fields of number theory and combinatorics, the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition.

Francis G. Garvan is an Australian-born mathematician who specializes in number theory and combinatorics. He holds the position Professor of Mathematics at the University of Florida. He received his Ph.D. from Pennsylvania State University with George E. Andrews as his thesis advisor. Garvan's thesis, Generalizations of Dyson's rank, concerned the rank of a partition and formed the groundwork for several of his later papers.

References

  1. Freeman J. Dyson (1944). "Some Guesses in The Theory of Partitions" (PDF). Eureka (Cambridge). 8: 10–15. ISBN   9780821805619.
  2. 1 2 3 George E. Andrews; F.G. Garvan (April 1988). "Dyson's crank of a partition" (PDF). Bulletin of the American Mathematical Society. New Series. 18 (2). Retrieved 26 November 2012.
  3. Srinivasa, Ramanujan (1919). "Some properties of p(n), number of partitions of n". Proceedings of the Cambridge Philosophical Society. XIX: 207–210.
  4. Manjil P. Saikia (2013). "Cranks in Ramanujan's Lost Notebook". Journal of the Assam Academy of Mathematics. 6. arXiv: 1402.6644 . Bibcode:2014arXiv1402.6644S.
  5. Manjil P. Saikia (2015). "A study of the crank function in Ramanujan's Lost Notebook". The Mathematics Student. 84. arXiv: 1406.3299 . Bibcode:2014arXiv1406.3299S.