The mex of a partition 
is the smallest positive
integer that does not occur as a part of 
. For example,
 |
and
 |
This differs by a shift of convention from the mex of a set of nonnegative integers.
The Frobenius-mex theorem states that the number of partitions of 
with even mex equals the number whose
Frobenius symbol has a zero in its top row. Odd mex corresponds to no zero in the
top row. The crank-mex theorem equates even and odd mex with negative and nonnegative
partition cranks, respectively. Lin et al.
(2026) give bijective refinements of both theorems
by the number of parts, 2-measure, Durfee square,
and Durfee rectangle.
