scholarly journals Congruence Properties of Binary Partition Functions

2013 ◽  
Vol 17 (1) ◽  
pp. 15-26 ◽  
Author(s):  
Katherine Anders ◽  
Melissa Dennison ◽  
Jennifer Weber Lansing ◽  
Bruce Reznick
Keyword(s):  
Partition Functions ◽  
Binary Partition ◽  
Congruence Properties

scholarly journals Congruence properties for a certain kind of partition functions

2016 ◽  
Vol 290 ◽  
pp. 739-772 ◽  
Author(s):  
Su-Ping Cui ◽  
Nancy S.S. Gu ◽  
Anthony X. Huang
Keyword(s):  
Partition Functions ◽  
Congruence Properties

scholarly journals Congruence properties ofS-partition functions

2011 ◽  
Vol 4 (4) ◽  
pp. 411-416
Author(s):  
Andrew Gruet ◽  
Linzhi Wang ◽  
Katherine Yu ◽  
Jiangang Zeng
Keyword(s):  
Partition Functions ◽  
Congruence Properties

On congruence properties of the restricted partition functions $$p_{\omega }(n,k)$$ p ω ( n , k ) and $$p_{\nu }(n,k)$$ p ν ( n , k )

2018 ◽  
Vol 49 (1) ◽  
pp. 105-113
Author(s):  
Robson da Silva ◽  
Kelvin Souza de Oliveira ◽  
Almir Cunha da Graça Neto
Keyword(s):  
Partition Functions ◽  
Restricted Partition ◽  
Congruence Properties

Some Binary Partition Functions

1990 ◽  
pp. 451-477 ◽  
Author(s):  
Bruce Reznick
Keyword(s):  
Partition Functions ◽  
Binary Partition

scholarly journals Some restricted partition functions: Congruences modulo 3

1970 ◽  
Vol 11 (1) ◽  
pp. 82-90 ◽  
Author(s):  
D. B. Lahiri
Keyword(s):  
Partition Functions ◽  
General Category ◽  
Previous Communication ◽  
Restricted Partition ◽  
Image Position ◽  
Congruence Properties

In a previous communication [5] the author has dealt with the congruence properties of some restricted partition functions. The general category of such functions may be denoted.

ON THE BROKEN 1-DIAMOND PARTITION

2008 ◽  
Vol 04 (02) ◽  
pp. 199-218 ◽  
Author(s):  
ERIC MORTENSON
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Infinite Family ◽  
Partition Functions ◽  
Congruence Properties

We introduce a crank-like statistic for a different class of partitions. In [4], Andrews and Paule initiated the study of broken k-diamond partitions. Their study of the respective generating functions led to an infinite family of modular forms, about which they were able to produce interesting arithmetic theorems and conjectures for the related partition functions. Here we establish a crank-like statistic for the broken 1-diamond partition and discuss its role in congruence properties.

Congruence properties of pk(n)

2019 ◽  
Vol 15 (06) ◽  
pp. 1267-1290 ◽  
Author(s):  
Julia Q. D. Du ◽  
Edward Y. S. Liu ◽  
Jack C. D. Zhao
Keyword(s):  
Positive Integer ◽  
Product Formula ◽  
Partition Functions ◽  
Euler Product ◽  
Modular Equations ◽  
Arbitrary Positive Integer ◽  
Modular Functions ◽  
Unified Approach ◽  
Linear Recurring Sequences ◽  
Congruence Properties

We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

Congruence properties of the binary partition function

1969 ◽  
Vol 66 (2) ◽  
pp. 371-376 ◽  
Author(s):  
R. F. Churchhouse
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Functional Equations ◽  
Precise Calculation ◽  
Binary Partition ◽  
Powers Of 2 ◽  
De Bruijn ◽  
Congruence Properties

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

Congruence properties of the binary partition function

1975 ◽  
Vol 78 (3) ◽  
pp. 437-442 ◽  
Author(s):  
M. D. Hirschhorn ◽  
J. H. Loxton
Keyword(s):  
Partition Function ◽  
Binary Partition ◽  
Powers Of 2 ◽  
Congruence Properties

We denote by b(n) the number of binary partitions of n, that is the number of partitions of n as the sum of powers of 2. As usual, two partitions are not considered to be distinct if they differ only in the order of their summands. It is convenient to define b(0) = 1.

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