Extending Andrews and Newman’s refinement of the Crank–Mex theorem
Volume 224 / 2026
Acta Arithmetica 224 (2026), 81-94
MSC: Primary 11P82; Secondary 05A17
DOI: 10.4064/aa251123-25-5
Published online: 18 June 2026
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 1. 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 1. We provide both analytic and combinatorial proofs.
