Function Repository Resource:

PartitionCrank

Get Dyson's crank of an integer partition

Contributed by: George Beck
 ResourceFunction["PartitionCrank"][x] gives Dyson's crank of the partition x.

Details and Options

A partition of a positive integer n is a weakly decreasing list of positive integers that add up to to n.
The crank of a partition is defined via a somewhat complicated piecewise function as discussed on the corresponding Wikipedia entry.
The crank is a partition statistic useful in studying congruence properties of the partition function p(n), given by PartitionsP in the Wolfram Language.

Examples

Basic Examples (2)

Here is a basic example:

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Here are the partitions of 9 in compact form:

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The crank of the partitions of 9:

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Neat Examples (3)

Ramanujan discovered and proved analytically that the number of partitions of 11m+6 is divisible by 11:

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Andrews and Garvan proved that the partitions of a number of the form 11m+6 can be split into 11 sets of equal size according to the crank mod 11:

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Here is an example using a different value of m:

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George Beck

Version History

• 1.0.0 – 07 June 2019