Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions.

Abstract

We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative integer m, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size n that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$ ), as n tends to infinity.