A Refinement of the Crank-Mex Theorem

TL;DR

This paper proves a new combinatorial identity linking partitions with odd mex and nonnegative crank, refining the Crank-Mex theorem.

Contribution

It establishes a refined equality between two classes of partitions, enhancing understanding of partition statistics.

Findings

Number of partitions with odd mex and non-one parts equals those with nonnegative crank and non-one parts.

Provides a combinatorial proof of the refined identity.

Deepens the connection between mex and crank statistics in partition theory.

Abstract

It is proved that the number of partitions of n with odd mex and k parts that aren't ones equals the number of partitions of n with nonnegative crank and k parts that aren't ones..