James A. Sellers

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A partition n = p1 + p2 + · · · + pk with 1 ≤ p1 ≤ p2 ≤ · · · ≤ pk is called non-squashing if p1 + · · · + pj ≤ pj+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 number of(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 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 = p1 + p2 + · · · + pk with 1 ≤ p1 ≤ p2 ≤ · · · ≤ pk wherein p1 + · · · + pj ≤ pj+1 for 1 ≤ j ≤ k − 1. Sloane has also hinted at a generalized box stacking problem which is closely related to generalized non–squashing(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 hyperbinary partition function (as they “recounted the rationals”). In this paper, we first prove an(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 b2(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)
Let bm(n) denote the number of partitions of n into powers of m. Define σr = ε2m 2 + ε3m 3 + · · · + εrm, where εi = 0 or 1 for each i. Moreover, let cr = 1 if m is odd, and cr = 2 r−1 if m is even. The main goal of this paper is to prove the congruence bm(m n− σr −m) ≡ 0 (mod m/cr). For σr = 0, the existence of such a congruence was conjectured by R. F.(More)