Crank of a partition

In 1988, George E. Andrews and Frank Garvan discovered a definition for the crank satisfying the properties hypothesized for it by Dyson.

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.

But the idea fails to divide partitions of integers of the form 11n + 6 into 11 classes of the same size, as the following table shows.

The generating function for M(m,n) is given below: Andrews and Garvan proved the following result[2] which shows that the crank as defined above does meet the conditions given by Dyson.

Recent work by Bruce C. Berndt and his coauthors argued that Ramanujan knew about the crank, although not in the form that Andrews and Garvan have defined.