Overpartitions in terms of 2-adic valuation

Abstract

In this paper, we consider the 2-adic valuation of integers and provide an alternative representation for the generating function of the number of overpartitions of an integer. As a consequence of this result, we obtain a new formula and a new combinatorial interpretation for the number of overpartitions of an integer. This formula implies a certain type of partitions with restrictions for which we provide two Ramanujan-type congruences and present as open problems two infinite families of linear inequalities. Connections between overpartitions and the game of m-Modular Nim with two heaps are presented in this context.