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- N. J. A. Sloane, James A. Sellers
- Discrete Mathematics
- 2005

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)

- James A. Sellers
- 2003

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 p1 + p2 + p3 + p4 + . . . such that p1 ≥ p2 ≥ p3 ≥ p4 ≥ · · · ≥ 0 and p1 ≥ 2p2 + p3 + p4 + . . . . Via partition analysis, we extend this result by replacing the last inequality with p1 ≥ k2p2+k3p3+k4p4+. . . , where k2,… (More)

In this article, we consider various arithmetic properties of the function po(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)

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)

1. Background and introduction. In his 1984 Memoir of the American Mathematical Society, Andrews [2] introduced two families of partition functions, φk(m) and cφk(m), which he called generalized Frobenius partition functions. In this paper, we will focus our attention on one of these functions, namely cφ2(m), which denotes the number of generalized… (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)

- George E. Andrews, James A. Sellers
- Discrete Mathematics
- 2007

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)

- Kevin M. Courtright, James A. Sellers
- 2004

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)

- Michael D. Hirschhorn, James A. Sellers
- Australasian J. Combinatorics
- 2004

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)

- James A. Sellers
- 2001

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)