CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS

CONGRUENCES MODULO SQUARES OF PRIMES FOR FU’S k DOTS BRACELET PARTITIONS

Abstract. In 2007, Andrews and Paule introduced the family of functions ∆k(n) which enumerate the number of broken k–diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, ∆1(2n + 1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized th...

Partition congruences by involutions

We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congruences. Introduction The theory of partitions is a beautiful subject introduced by Euler over 250 years ago and is still under intense development [2]. Arguably, a turning point in its history was the invention of the “constructive partition theory” symbolized by F...

Square - Partition Congruences

Congruences of the Partition Function

Ramanujan also conjectured that congruences (1) exist for the cases A = 5 , 7 , or 11 . This conjecture was proved by Watson [17] for the cases of powers of 5 and 7 and Atkin [3] for the cases of powers of 11. Since then, the problem of finding more examples of such congruences has attracted a great deal of attention. However, Ramanujan-type congruences appear to be very sparse. Prior to the la...

Ramanujan Type Congruences for a Partition Function

We investigate the arithmetic properties of a certain function b(n) given by ∞ ∑ n=0 b(n)qn = (q; q)−2 ∞ (q 2; q2)−2 ∞ . One of our main results is b(9n + 7) ≡ 0 (mod 9).