scholarly journals Dancing samba with Ramanujan partition congruences

2018 ◽  
Vol 84 ◽  
pp. 14-24 ◽  
Author(s):  
Ralf Hemmecke
Keyword(s):  
Partition Congruences

scholarly journals PARTITION CONGRUENCES AND DYSON’S RANK

2014 ◽  
Vol 2 (2) ◽  
pp. 49-60
Author(s):  
Sabuj Das
Keyword(s):  
Generating Functions ◽  
Modular Functions ◽  
Partition Congruences ◽  
Freeman Dyson

In this article the rank of a partition of an integer is a certain integer associated with the partition. The term has first introduced by freeman Dyson in a paper published in Eureka in 1944. In 1944, F.S. Dyson discussed his conjectures related to the partitions empirically some Ramanujan’s famous partition congruences. In 1921, S. Ramanujan proved his famous partition congruences: The number of partitions of numbers 5n+4, 7n+5 and 11n +6 are divisible by 5, 7 and 11 respectively in another way. In 1944, Dyson defined the relations related to the rank of partitions. These are later proved by Atkin and Swinnerton-Dyer in 1954. The proofs are analytic relying heavily on the properties of modular functions. This paper shows how to generate the generating functions for In this paper, we show how to prove the Dyson’s conjectures with rank of partitions.

scholarly journals Partition Statistics for Cubic Partition Pairs

10.37236/615 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Byungchan Kim
Keyword(s):  
Partition Congruences ◽  
Partition Statistics

In this brief note, we give two partition statistics which explain the following partition congruences: \begin{align*} b(5n+4) &\equiv 0 \pmod{5}, \\ b(7n+a) &\equiv 0 \pmod{7}, \text{if $a=2$, $3$, $4$, or $6$}. \end{align*} Here, $b(n)$ is the number of $4$-color partitions of $n$ with colors $r$, $y$, $o$, and $b$ subject to the restriction that the colors $o$ and $b$ appear only in even parts.

SUMS OF SQUARES AND PARTITION CONGRUENCES

2020 ◽  
pp. 1-19
Author(s):  
SU-PING CUI ◽  
NANCY S. S. GU
Keyword(s):  
Positive Integer ◽  
Sums Of Squares ◽  
Binomial Coefficients ◽  
Partition Congruences ◽  
Arithmetic Properties ◽  
Positive Integers ◽  
Pair Function

For positive integers $n$ and $k$ , let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$ , by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$ . Furthermore, in view of these arithmetic properties of $r_{k}(n)$ , we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

scholarly journals A simple proof of Watson's partition congruences for powers of 7

1984 ◽  
Vol 36 (3) ◽  
pp. 316-334 ◽  
Author(s):  
F. G. Garvan
Keyword(s):  
Simple Proof ◽  
High Power ◽  
Specific Form ◽  
Modular Equation ◽  
Partition Congruences ◽  
Seventh Order

AbstractRamanujan conjectured that if n is of a specific form then p(n), the number of unrestricted partitions of n, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson.In this paper we establish appropriate generating formulae, from which Watson's results follow easily.Our proofs are more straightforward than those of Watson. They are elementary, depending only on classical identities of Euler and Jacobi. Watson's proofs rely on the modular equation of seventh order. We also need the modular equation but we derive it using the elementary techniques of O. Kolberg.

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