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.

中文翻译：

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戴森的曲柄和整数分区的 mex

Andrews 和 Newman 最近引入了分区的 mex 的概念，即不是部分的最小正整数。不过，这个概念至少从 2006 年就开始使用，与 Frobenius 符号有关。最近，mex 的奇偶性与 Dyson 于 1944 年命名的曲柄统计量相关联。在本说明中，我们通过证明许多与这些分区统计数据自然相关的属性。