Dyson's crank and the mex of integer partitions

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2006, though, with connections to Frobenius symbols. Recently, the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generalization of the mex) by proving a number of properties that naturally relate these partition statistics.