Creep and shrinkage of concrete

The rank of a partition, shown as its Young diagram

In mathematics, in theory of numbers, the rank of a partition of a positive integer is a certain integer associated with the partition of the integer. 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 published annually by the Mathematical Society of Cambridge University. 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.

Definition

By a partition of a positive integer n we mean a finite set of integers λ = { λ_k, λ_{k − 1}, . . . , λ₁ } satisfying the following two conditions:

• λ_k ≥ . . . ≥ λ₂ ≥ λ₁ > 0.
• λ_k + . . . + λ₂ + λ₁ = n.

If λ_k, . . . , λ₂, λ₁ are distinct, that is, if

• λ_k > . . . > λ₂ > λ₁ > 0

the partition λ is called a strict partition of n. The integers λ_k, λ_{k − 1}, ..., λ₁ are the parts of the partition. The number of parts in the partition λ is k and the largest part in the partition is λ_k. The rank of the partition λ (whether ordinary or strict) is defined as λ_k − k.

The ranks of the partitions of n take the following values and no others:

n − 1, n −3, n −4, . . . , 2, 1, 0, −1, −2, . . . , −(n − 4), −(n − 3), −(n − 1).

The following table gives the ranks of the various partitions of the number 5.

Ranks of the partitions of the integer 5

Partition (λ)	Largest part (λk)	Number of parts (k)	Rank of the partition (λk − k )
{ 5 }	5	1	4
{ 4, 1 }	4	2	2
{ 3, 2 }	3	2	1
{ 3, 1, 1 }	3	3	0
{ 2, 2, 1 }	2	3	−1
{ 2, 1, 1, 1 }	2	4	−2
{ 1, 1, 1, 1, 1 }	1	5	−4

Notations

The following notations are used to specify how many partitions have a given rank. Let n, q be a positive integers and m be any integer.

• The total number of partitions of n is denoted by p(n).
• The number of partitions of n with rank m is denoted by N(m, n).
• The number of partitions of n with rank congruent to m modulo q is denoted by N(m, q, n).
• The number of strict partitions of n is denoted by Q(n).
• The number of strict partitions of n with rank m is denoted by R(m, n).
• The number of strict partitions of n with rank congruent to m modulo q is denoted by T(m, q, n).

For example,

p(5) = 7 , N(2, 5) = 1 , N(3, 5) = 0 , N(2, 2, 5) = 5 .

Q(5) = 3 , R(2, 5) = 1 , R(3, 5) = 0 , T(2, 2, 5) = 2.

Some basic results

Let n, q be a positive integers and m be any integer.

• $N(m,n)=N(-m,n)$
• $N(m,q,n)=N(q-m,q,n)$
• $N(m,q,n)=\sum _{r=-\infty }^{\infty }N(m+rq,n)$

Ramanujan's congruences and Dyson's conjecture

Srinivasa Ramanujan in a paper published in 1919 proved the following congruences involving the partition function p(n):

• p(5 n + 4) ≡ 0 (mod 5)
• p(7n + 5) ≡ 0 (mod 7)
• p(11p + 6) ≡ 0 (mod 11)

In commenting on this result, Dyson noted that " . . . although we can prove that the partitions of 5n + 4 can be divided into five equally numerous subclasses, it is unsatisfactory to receive from the proofs no concrete idea of how the division is to be made. We require a proof which will not appeal to generating functions, . . . ". Dyson introduced the idea of rank of a partition to accomplish the task he set for himself. Using this new idea, he made the following conjectures:

• N(0, 5, 5n + 4) = N(1, 5, 5n + 4) = N(2, 5, 5n + 4) = N(3, 5, 5n + 4) = N(4, 5, 5n + 4)
• N(0, 7, 7n + 5) = N(1, 7, 7n + 5) = N(2, 7, 7n + 5) = . . . = N(6, 7, 7n + 5)

These conjectures were proved by Atkin and Swinnerton-Dyer in 1954.

The following tables show how the partitions of the integers 4 (5 × n + 4 with n = 0) and 9 (5 × n + 4 with n = 1 ) get divided into five equally numerous subclasses.

Partitions of the integer 4

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)
{ 2, 2 }	{ 3, 1 }	{ 1, 1, 1, 1 }	{ 4 }	{ 2, 1, 1 }

Partitions of the integer 9

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 }

Generating functions

• The generating function of p(n) was discovered by Euler and is well-known.

$\sum _{n=0}^{\infty }p(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{(1-x^{k})}}$

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

$\sum _{m=-\infty }^{\infty }\sum _{n=0}^{\infty }N(m,n)z^{m}q^{n}=1+\sum _{n=1}^{\infty }{\frac {q^{n^{2}}}{\prod _{k=1}^{n}(1-zq^{k})(1-z^{-1}q^{k})}}$

• The generating function for Q ( n ) is given below:

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

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

$\sum _{m,n=0}^{\infty }Q(m,n)z^{m}q^{n}=1+\sum _{s=1}^{\infty }{\frac {q^{s(s+1)/2}}{(1-zq)(1-zq^{2})\cdots (1-zq^{s})}}$

Further reading

• Asymptotic formulas for the rank partition function:

• Congruences for rank function:

• Generalisation of rank to BG-rank:

See also

Crank of a partition

References

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