Rank of a partition

Definition

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

• λk ≥ . . . ≥ λ2 ≥ λ1 0.
• *λ*k + . . . + λ2 + λ1 = n. If *λ*k, . . . , *λ*2, *λ*1 are distinct, that is, if
• *λ*k . . . λ2 λ1 0 then the partition λ is called a strict partition of n. The integers *λ*k, λk − 1, ..., *λ*1 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

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(5n + 4) ≡ 0 (mod 5)
• p(7n + 5) ≡ 0 (mod 7)
• p(11n + 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 of the integer 9

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 R(m, n) is given below:

:: \sum_{m,n=0}^\infty R(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)}

Alternate definition

Main article: Durfee square

In combinatorics, the phrase rank of a partition is sometimes used to describe a different concept: the rank of a partition λ is the largest integer i such that λ has at least i parts each of which is no smaller than i. Equivalently, this is the length of the main diagonal in the Young diagram or Ferrers diagram for λ, or the side-length of the Durfee square of λ.

The table of ranks (under this alternate definition) of partitions of 5 is given below.

Ranks of the partitions of the integer 5

References

References

1. F. Dyson. (1944). "Some guesses in the theory of partitions". Eureka (Cambridge).
2. Srinivasa, Ramanujan. (1919). "Some properties of ''p''(''n''), number of partitions of ''n''". Proceedings of the Cambridge Philosophical Society.
3. A. O. L. Atkin. (1954). "Some properties of partitions". Proceedings of the London Mathematical Society.
4. G.H. Hardy and E.W. Wright. (1938). "An introduction to the theory of numbers". Oxford University Press.
5. Bringmann, Kathrin. (2009). "Congruences for Dyson's ranks". International Journal of Number Theory.
6. Maria Monks. (2010). "Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts". [[Proceedings of the American Mathematical Society]].
7. [[Richard P. Stanley. Stanley, Richard P.]] (1999) [http://www-math.mit.edu/~rstan/ec/ ''Enumerative Combinatorics'', Volume 2], p. 289. [[Cambridge University Press]]. {{ISBN. 0-521-56069-1.
8. Bringman, Kathrin. (July 2009). "Asymptotics For Rank Partition Functions". Transactions of the American Mathematical Society.
9. Bringmann, Kathrin. (2009). "Congruences for Dyson's rank". International Journal of Number Theory.
10. (2008). "The BG-rank of a partition and its applications". Advances in Applied Mathematics.