Relating the crank of a partition and smallest missing parts

Abstract: The primary goal of this talk is to demonstrate a natural connection between the smallest missing part of an integer partition (commonly referred to as the ``mex" of the partition) and the concept of the crank of a partition. After providing a brief history of the crank of a partition a la Dyson as well as Andrews and Garvan, we will utilize straightforward generating function manipulations to make this connection. We will then consider additional results on the mex statistic based on parity, and we will also demonstrate connections between the crank and Frobenius symbols which satisfy certain conditions.

This work is joint with Brian Hopkins, Dennis Stanton, and Ae Ja Yee.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)