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Concept
Crank of a partition
Summary
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.
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 published in 1918 stated and proved the following congruences for the partition function p(n), since known as Ramanujan congruences.
p(5n + 4) ≡ 0 (mod 5)
p(7n + 5) ≡ 0 (mod 7)
p(11n + 6) ≡ 0 (mod 11)
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:
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).
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