For a partition 
, let 
be the number of parts equal to 1, let 
be the number of parts larger than 
, and let 
be its largest part. The crank of 
is
 |
(1)
|
For example, the crank of 
is 7, while the crank of 
is 
.
Dyson introduced the crank to explain Ramanujan's partition congruences, and Andrews and Garvan (1988) found the statistic above. The crank-mex
theorem states that the number of partitions of 
with negative crank equals the number
with even partition mex; equivalently, nonnegative
crank corresponds in number to odd partition mex.
Lin et al. (2026) give bijective refinements
preserving the number of nonunit parts and the 2-measure.
