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ARITHMETIC PROPERTIES OF PARTITION QUADRUPLES WITH ODD PARTS DISTINCT

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  08 July 2015

LIUQUAN WANG
LIUQUAN WANG*
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore email [email protected], [email protected]
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Abstract

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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 $n\geq 0$,

$$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$
We also establish some internal congruences and some Ramanujan-type congruences modulo 2, 5 and 8 satisfied by $\text{pod}_{-4}(n)$.

Keywords

congruencespartition quadruplesdistinct odd partstheta functions

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 353 - 364
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

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