Crank of a partition
In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The term was first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal published by the Mathematics Society of Cambridge University. Dyson then 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.
Dyson's crank
Let n be a non-negative integer and let p denote the number of partitions of n. Srinivasa Ramanujan in a paper published in 1918 stated and proved the following congruences for the partition function p, since known as Ramanujan congruences.- p ≡ 0
- p ≡ 0
- p ≡ 0
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 = of 9 is 4 − 5 = −1. Denoting by N, the number of partitions of n whose ranks are congruent to m modulo q, Dyson considered N and N 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:
- N = N = N = N = N.
- N = N = N = N = N = N = N
Partitions of the integer 6 divided into classes based on ranks
| rank ≡ 0 | rank ≡ 1 | rank ≡ 2 | rank ≡ 3 | rank ≡ 4 | rank ≡ 5 | rank ≡ 6 | rank ≡ 7 | rank ≡ 8 | rank ≡ 9 | rank ≡ 10 |
Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,
I hold in fact :
- that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall call this hypothetical coefficient the "crank" of the partition and denote by M the number of partitions of n whose crank is congruent to m modulo q;
- that M = M;
- that M = M = M = M = M;
- that...
Definition of crank
In a paper published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows: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 ℓ | Number of 1's ω | Number of parts larger than ω μ | Crank c |
| 4 | 0 | 1 | 4 | |
| 3 | 1 | 1 | 0 | |
| 2 | 0 | 2 | 2 | |
| 2 | 2 | 0 | −2 | |
| 1 | 4 | 0 | −4 |
Cranks of the partitions of 5
| Partition λ | Largest part ℓ | Number of 1's ω | Number of parts larger than ω μ | Crank c |
| 5 | 0 | 1 | 5 | |
| 4 | 1 | 1 | 0 | |
| 3 | 0 | 2 | 3 | |
| 3 | 2 | 1 | −1 | |
| 2 | 1 | 2 | 1 | |
| 2 | 3 | 0 | −3 | |
| 1 | 5 | 0 | −5 |
Cranks of the partitions of 6
| Partition λ | Largest part ℓ | Number of 1's ω | Number of parts larger than ω μ | Crank c |
| 6 | 0 | 1 | 6 | |
| 5 | 1 | 1 | 0 | |
| 4 | 0 | 2 | 4 | |
| 4 | 2 | 1 | −1 | |
| 3 | 0 | 2 | 3 | |
| 3 | 1 | 2 | 1 | |
| 3 | 3 | 0 | −3 | |
| 2 | 0 | 3 | 2 | |
| 2 | 2 | 0 | −2 | |
| 2 | 4 | 0 | −4 | |
| 1 | 6 | 0 | −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 except for n = 1 where M = −M = M = 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.The generating function for M is given below:
Basic result
Andrews and Garvan proved the following result which shows that the crank as defined above does meet the conditions given by Dyson.- M = M = M = M = M = p / 5
- M = M = M = M = M = M = M = p / 7
- M = M = M = M =... = M = M = p / 11
Classification of the partitions of the integer 9 based on cranks
| Partitions with crank ≡ 0 | Partitions with crank ≡ 1 | Partitions with crank ≡ 2 | Partitions with crank ≡ 3 | Partitions with crank ≡ 4 |
Classification of the partitions of the integer 9 based on ranks
| Partitions with rank ≡ 0 | Partitions with rank ≡ 1 | Partitions with rank ≡ 2 | Partitions with rank ≡ 3 | Partitions with rank ≡ 4 |