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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 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)
Numerous functions which enumerate partitions into powers of a fixed number m have been studied ever since Churchhouse's original work in the late 1960's on the unrestricted binary partition function. In particular, Calkin and Wilf recently considered the hyper-binary partition function (as they " recounted the rationals "). In this paper, we first prove an… (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)
Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D n (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are… (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 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)
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 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)
Let b m (n) denote the number of partitions of n into powers of m.