The Frobenius-Mex Theorem and the Crank-Mex Theorem

Abstract

The mex, as a relatively new partition statistic, has attracted great attention over the past five years and has led to a number of interesting developments. The Frobenius-Mex theorem and the Crank-Mex theorem stand out as two of the most significant results in this area. In this paper, we not only further refine these two theorems in distinct ways, but also provide bijective proofs for all of our refinements. Notably, our work answers Andrews and Newman's request for a bijection underlying their refinement of the Crank-Mex theorem.