On m-Ary Overpartitions

Presently there are a lot of activities in the study of overpartitions, objects that were discussed by MacMahon, and which have recently proven useful in several combinatorial studies of basic hypergeometric series. In this paper we study some similar objects, which we name m-ary overpartitions. We consider divisibility properties of the number of m-ary overpartitions of a natural number, and we prove a theorem which is a lifting to general m of the well-known Churchhouse congruences for the binary partition function. Received October 11, 2004     

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

Overpartitions and ternary quadratic forms

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, Mahlburg showed that \(\overline{p}(n) \equiv 0 \pmod {64}\) and Kim showed that \(\overline{p}(n) \equiv 0 \pmod {128}\) for almost all integers n. In this paper, with the help of some ternary quadratic forms, we prove that \(\overline{p}(n) \equiv 0 \pmod {256}\) for almost all integers n, which was conjectured by Mahlburg.     

Some Congruences for Overpartitions with Restriction

Mathematical Notes - For any given positive integersmand n, let pm(n) denote the number of overpartitions of n with no parts divisible by 4mand only the parts congruent tommodulo 2moverlined. In...     

Asymptotics of Higher Order Ospt-Functions for Overpartitions

In this paper we obtain asymptotic formulas for positive crank and rank moments for overpartitions. Moreover, we show that crank and rank moments are asymptotically equal while the difference is asymptotically positive. This indicates that there exist analogous higher ospt-functions for overpartitions, which we define.     

Arithmetic Properties of Overpartitions into Odd Parts

In this article, we consider various arithmetic properties of the function which denotes the number of overpartitions of n using only odd parts. This function has arisen in a number of recent papers, but in contexts which are very different from overpartitions. We prove a number of arithmetic results including several Ramanujan-like congruences satisfied by and some easily-stated characterizations of modulo small powers of two. For example, it is proven that, for n ≥ 1, (mod 4) if and only if n is neither a square nor twice a square. Received March 17, 2005     

Multiplicity and Number of Parts in Overpartitions

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 .     

The First Positive Rank and Crank Moments for Overpartitions

In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.     

Overpartitions, lattice paths, and Rogers-Ramanujan identities

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers-Ramanujan type. Among these are Gordon's generalization of the Rogers-Ramanujan identities, Andrews' generalization of the Göllnitz-Gordon identities, and Lovejoy's “Gordon's theorems for overpartitions.”