Rank of a partition

In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer 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.[1] 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.

The rank of a partition, shown as its Young diagram
Freeman Dyson in 2005

. . . Rank of a partition . . .

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 . . . λ2λ1 > 0.
  • λk + . . . + λ2 + λ1 = n.

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

  • λk > . . . > λ2 > λ1 > 0

then the partition λ is called a strict partition of n. The integers λk, λk 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 λk. The rank of the partition λ (whether ordinary or strict) is defined as λkk.[1]

The ranks of the partitions of n take the following values and no others:[1]

n 1, n3, n4, . . . , 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
(λkk )
{ 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

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.

. . . Rank of a partition . . .

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. . . Rank of a partition . . .