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

We prove specific biases in the number of occurrences of parts belonging to two different residue classes <i>a</i> and <i>b</i>, modulo a fixed nonnegative integer <i>m</i>, 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 <i>n</i> that belong to these sets of partitions and have a symmetric residue class bias (i.e., for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>a</mml:mi> <mml:mo><</mml:mo> <mml:mi>m</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mi>a</mml:mi></mml:mrow> </mml:math> ), as <i>n</i> tends to infinity.