Abstract
Broken \(k\)-diamond partitions were introduced in 2007 by Andrews and Paule. Let \(\Delta _k(n)\) denote the number of broken \(k\)-diamond partitions of \(n\). In 2010, Radu and Sellers provided many beautiful congruences for \(\Delta _k(n)\) modulo 2 when \(k=2,3,5,6,8,9,11\). Among them when \(k=8\), they showed that \(\Delta _8(34n+r)\equiv ~0\pmod {2}\) when \(r\in \{11,15,17,19,25,27,29,33\}\). In this article, by using properties of modular forms, we extend this result for \(\Delta _8(n)\). We have completely determined the behavior of \(\Delta _8(2n+1)\) modulo 2. As a consequence, we obtain many more congruences for \(\Delta _8(n)\) modulo 2.
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The author is grateful to the anonymous referee for the useful suggestions.
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This work was supported by the National Nature Science Foundation of China (Grant No. 11071160).
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Wang, Y. More parity results for broken 8-diamond partitions. Ramanujan J 39, 339–346 (2016). https://doi.org/10.1007/s11139-014-9660-x
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DOI: https://doi.org/10.1007/s11139-014-9660-x