Binary Partitions Revisited

Rødseth, Øystein J.; Sellers, James A.

doi:10.1006/jcta.2001.3223

Cited by 10 publications

(9 citation statements)

References 5 publications

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“…(A similar family of congruences for a restricted binary partition function was also proved by Rødseth and Sellers [2].) With the above in mind, we can shed more direct light on our goal for this note by first proving the following two infinite families of divisibility properties modulo 2 satisfied by b(n).…”

Section: Theorem 2 ([5] Theorem 2) the Generating Function For B(n) supporting

confidence: 52%

Infinite Families of Divisibility Properties Modulo 4 for Non-Squashing Partitions into Distinct Parts

Hirschhorn¹,

Rødseth²,

Sellers³

2009

Integers

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In 2005, Sloane and Sellers defined a function b(n) which denotes the number of nonsquashing partitions of n into distinct parts. In their 2005 paper, Sloane and Sellers also proved various congruence properties modulo 2 satisfied by b(n). In this note, we extend their results by proving two infinite families of congruence properties modulo 4 for b(n). In particular, we prove that for all k ≥ 3 and all n ≥ 0, b(2 2k+1 n + 2 2k−2 ) ≡ 0 (mod 4) and b(2 2k+1 n + 3 · 2 2k−2 + 1) ≡ 0 (mod 4).

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Section: Theorem 2 ([5] Theorem 2) the Generating Function For B(n) supporting

confidence: 52%

Infinite Families of Divisibility Properties Modulo 4 for Non-Squashing Partitions into Distinct Parts

Hirschhorn¹,

Rødseth²,

Sellers³

2009

Integers

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“…For m = 2, this is essentially Theorem 3.6 in Reznick [12]. In a series of papers (see [13,14] and the references therein) it has been shown that b m,k (n) possesses certain divisibility properties. From Section 3 we now get divisibility properties of a rather different type.…”

Section: A Special Case: M-ary Partitionsmentioning

confidence: 91%

Partitions With Parts in a Finite Set

Rødseth

Sellers

2006

Int. J. Number Theory

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For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.

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“…A number of proofs of (1.2) have been given by several authors; cf. [4]. Families of congruences also appear in the literature for the m-ary partition function which are valid for any m ≥ 2; cf.…”

Section: Introductionmentioning

confidence: 99%

On m-Ary Overpartitions

Rødseth

Sellers

2005

Ann. Comb.

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Presently there are a lot of activities in the study of overpartitions, objects that were discussed by MacMahon, and which have recently proven useful in several combinatorial studies of basic hypergeometric series. In this paper we study some similar objects, which we name m-ary overpartitions. We consider divisibility properties of the number of m-ary overpartitions of a natural number, and we prove a theorem which is a lifting to general m of the well-known Churchhouse congruences for the binary partition function.

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Binary Partitions Revisited

Abstract: The restricted binary partition function b k (n) enumerates the number of ways to represent n as n=2We study the question of how large a power of 2 divides the difference b k (2 r+2 n) − b k − 2 (2 r n) for fixed k \ 3, r \ 1, and all n \ 1.

Cited by 10 publications

References 5 publications

Infinite Families of Divisibility Properties Modulo 4 for Non-Squashing Partitions into Distinct Parts

Infinite Families of Divisibility Properties Modulo 4 for Non-Squashing Partitions into Distinct Parts

Partitions With Parts in a Finite Set

On m-Ary Overpartitions

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