The Cranks for 5-Core Partitions

It is well known that the number of 5-core partitions of 5 kn + 5 k ？ 1 is a multiple of 5 k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5 n + 4 and 25 n + 24 into 5 and 25 classes of equal size, respectively. In this paper we will develop the cranks that can be used to sort the 5-core partitions of 5 kn + 5 k ？ 1 into 5 k classes of equal size.