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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 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 1969, R. F. Churchhouse [2] 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 [1], Gupta [4–6], and Rodseth [7] extended Churchhouse's results. They considered a(More)
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)
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)