A partition n = p 1 + p 2 + · · · + p k with 1 ≤ p 1 ≤ p 2 ≤ · · · ≤ p k is called non-squashing if p 1 + · · · + p j ≤ p j+1 for 1 ≤ j ≤ k − 1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the… (More)
In 1969, R. F. Churchhouse  studied the number of binary partitions of an integer n. That is, Churchhouse proved various properties of the partition function b 2 (n), which counts the number of partitions of n into parts which are powers of 2. Soon after, Andrews , Gupta [4–6], and Rodseth  extended Churchhouse's results. They considered a… (More)
In this work, we consider the function ped(n), the number of partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan's congruences for the unrestricted partition function p(n). We prove a number of results for… (More)
In this article, we consider various arithmetic properties of the function p o (n) 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… (More)
γνωσι πoλλoυς αι µπνoιης The goal of this paper is to prove two new congruences involving 4–cores using elementary techniques; namely, if a 4 (n) denotes the number of 4–cores of n, then a 4 (9n+2) ≡ 0 (mod 2) and a 4 (9n + 8) ≡ 0 (mod 4).
In this work, we consider the function pod(n), the number of partitions of an integer n wherein the odd parts are distinct (and the even parts are unrestricted), a function which has arisen in recent work of Alladi. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan's congruences for the unrestricted partition… (More)
In a recent note, Santos proved that the number of partitions of n using only odd parts equals the number of partitions of n of the form p 1 + p 2 + p 3 + p 4 +. .. such that p 1 ≥ p 2 ≥ p 3 ≥ p 4 ≥ · · · ≥ 0 and p 1 ≥ 2p 2 + p 3 + p 4 +. .. . Via partition analysis, we extend this result by replacing the last inequality with p 1 ≥ k 2 p 2 +k 3 p 3 +k 4 p 4… (More)
A number of arithmetic properties of overpartitions have been proven recently. However, all such results have involved moduli which are powers of 2. In this brief note, we prove the first infinite family of congruences with a modulus that is not a power of 2 by proving that, for all n ≥ 0 and all α ≥ 0, p(9 α (27n + 18)) ≡ 0 (mod 12).
Recently, Sloane and Sellers solved a certain box stacking problem related to non– squashing partitions. These are defined as partitions n = p 1 + p 2 + · · · + p k with 1 ≤ p 1 ≤ p 2 ≤ · · · ≤ p k wherein p 1 + · · · + p j ≤ p j+1 for 1 ≤ j ≤ k − 1. Sloane has also hinted at a generalized box stacking problem which is closely related to generalized… (More)
In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell… (More)