Extending Andrews and Newman's refinement of the crank-mex theorem

TL;DR

This paper extends the crank-mex theorem by establishing new refined relationships between partitions with even mex, fixed points, and crank values, supported by analytic and combinatorial proofs.

Contribution

It introduces a complementary refinement connecting even mex partitions with negative and positive crank, expanding upon Andrews and Newman's original theorem.

Findings

Established a new refined relationship involving even mex and crank values.

Provided both analytic and combinatorial proofs of the new results.

Expanded the understanding of partition statistics related to crank and mex.

Abstract

The crank-mex theorem states that the number of integer partitions of $n$ with nonnegative crank equals the number with odd minimal excludant (mex). Andrews and M. Newman recently refined that result in terms of the number of parts greater than one. Here, we establish and expand a complementary result connecting the partitions with even mex, having fixed points, with negative crank, and with positive crank, all refined in terms of number of parts greater than one. We provide both analytic and combinatorial proofs.