Multiplicity and Number of Parts in Overpartitions

摘要

The purpose of this paper is to study the parts, part sizes and multiplicities in overpartitions using combinatorics, probabilities and asymptotics. We show that the probability that a randomly chosen part size of a randomly chosen overpartition of n has multiplicity m or m + 1 approaches 1/(m(m + 1) ln 2) and that the expected multiplicity of a randomly chosen part size of a randomly chosen overpartition of n approaches ln n/(4ln 2) as n→ ∞.