Dyson's crank and the mex of integer partitions (Q2237935)

The partition crank function was developed by \textit{G. E. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167--171 (1988; Zbl 0646.10008)] and \textit{F. G. Garvan} [Trans. Am. Math. Soc. 305, No. 1, 47--77 (1988; Zbl 0641.10009)] with the goal of giving a combinatorial proof of Ramanujan's congruence \(p(11n+6)\equiv 0\pmod {11}\). The notion of the mex of a partition, the smallest positive integer that is not a part, have been recently considered by \textit{G. E. Andrews} and \textit{D. Newman} [Ann. Comb. 23, No. 2, 249--254 (2019; Zbl 1458.11011)]. In this paper, the authors introduce a generalization of the mex statistic and connect it to the partitions with a given minimum crank. A lot of properties that naturally link these partition statistics are provided in this context.