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CONGRUENCES MODULO POWERS OF 2 FOR FU’S 5 DOTS BRACELET PARTITIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 22 August 2013, pp. 360-372
- Print publication:
- June 2014
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- Article
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In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken
$k$-diamond partitions. Recently, Shishuo Fu generalised the notion of broken 
$k$-diamond partitions to combinatorial objects which he termed 
$k$ dots bracelet partitions. Fu denoted the number of 
$k$ dots bracelet partitions of 
$n$ by 
${\mathfrak{B}}_{k} (n)$ and proved several congruences modulo primes and modulo powers of 2. More recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 
${\mathfrak{B}}_{5} (n)$, 
${\mathfrak{B}}_{7} (n)$ and 
${\mathfrak{B}}_{11} (n)$. In this note, we prove some congruences modulo powers of 2 for 
${\mathfrak{B}}_{5} (n)$. For example, we find that for all integers 
$n\geq 0$, 
${\mathfrak{B}}_{5} (16n+ 7)\equiv 0\hspace{0.167em} ({\rm mod} \hspace{0.334em} {2}^{5} )$.
