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ARITHMETIC PROPERTIES OF PARTITION QUADRUPLES WITH ODD PARTS DISTINCT
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 3 / December 2015
- Published online by Cambridge University Press:
- 08 July 2015, pp. 353-364
- Print publication:
- December 2015
-
- Article
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- You have access
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Let
$\text{pod}_{-4}(n)$ denote the number of partition quadruples of 
$n$ where the odd parts in each partition are distinct. We find many arithmetic properties of 
$\text{pod}_{-4}(n)$ including the following infinite family of congruences: for any integers 
${\it\alpha}\geq 1$ and 
$n\geq 0$, We also establish some internal congruences and some Ramanujan-type congruences modulo 2, 5 and 8 satisfied by
$$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$
$\text{pod}_{-4}(n)$.
