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Congruences modulo 16, 32, and 64 for Andrews’s singular overpartitions

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Abstract

In a recent work, Andrews gave a definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function \(\overline{C}_{k,i}(n) \) which denotes the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i \ (\mathrm{mod}\ k)\) may be overlined. Andrews, Chen, Hirschhorn and Sellers, and Ahmed and Baruah discovered numerous congruences modulo 2, 3, 4, 8, and 9 for \(\overline{C}_{3,1}(n)\). In this paper, we prove a number of congruences modulo 16, 32, and 64 for \(\overline{C}_{3,1}(n)\).

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Acknowledgments

The author would like to thank the anonymous referee for valuable suggestions which resulted in a great improvement of the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Olivia X. M. Yao

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  1. Olivia X. M. Yao
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Correspondence to Olivia X. M. Yao.

Additional information

This work was supported by the National Natural Science Foundation of China (11401260 and 11571143).

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Yao, O. .M. Congruences modulo 16, 32, and 64 for Andrews’s singular overpartitions. Ramanujan J 43, 215–228 (2017). https://doi.org/10.1007/s11139-015-9760-2

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  • DOI: https://doi.org/10.1007/s11139-015-9760-2

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