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Congruences modulo 7 for broken 3-diamond partitions

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Abstract

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Wang and Yao, and Xia proved several infinite families of congruences modulo 7 for \(\Delta _3(n)\) by using theta function identities. In this paper, we give a new proof of one result of Wang and Yao, and find three new infinite families of congruences modulo 7 for \(\Delta _3(n)\).

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References

  1. Andrews, G.E., Paule, P.: MacMahon’s partition analysis XI: broken diamonds and modular forms. Acta Arithm. 126, 281–294 (2007)

    Article  MathSciNet  Google Scholar 

  2. Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)

    Book  Google Scholar 

  3. Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006)

    Book  Google Scholar 

  4. Jameson, M.: Congruences for broken \(k\)-diamond partitions. Ann. Combin. 17, 333–338 (2013)

    Article  MathSciNet  Google Scholar 

  5. Paule, P., Radu, S.: Infinite families of strange partition congruences for broken 2-diamonds. Ramanujan J. 23, 409–416 (2010)

    Article  MathSciNet  Google Scholar 

  6. Xia, E.X.W.: Infinite families of congruences modulo 7 for broken 3-diamond partitions. Ramanujan J. 40, 389–403 (2016)

    Article  MathSciNet  Google Scholar 

  7. Xia, E.X.W.: More infinite families of congruences modulo 5 for broken 2-diamond partitions. J. Number Theory 170, 250–262 (2017)

    Article  MathSciNet  Google Scholar 

  8. Xia, E.X.W.: Congruences modulo 9 and 27 for overpartitions. Ramanujan J. 42, 301–323 (2017)

    Article  MathSciNet  Google Scholar 

  9. Xiong, X.: Two congruences involving Andrews–Paule’s broken \(3\)-diamond partitions and 5-diamond partitions. Proc. Jpn. Acad. Ser. A (Math. Sci.) 87, 65–68 (2011)

    Article  MathSciNet  Google Scholar 

  10. Yao, O.X.M., Wang, Y.J.: Newman’s identity and infinite families of congruences modulo 7 for broken 3-diamond partitions. Ramanujan J. 43, 619–631 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, 201620, People’s Republic of China

    Eric H. Liu

  2. Wenbo College, East China University of Political Science and Law, Shanghai, 201620, People’s Republic of China

    Wenjing Du

Authors
  1. Eric H. Liu
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  2. Wenjing Du
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Corresponding author

Correspondence to Eric H. Liu.

Additional information

This work was supported by the National Science Foundation of China (Grant Nos. 11701362 and 11501356).

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Liu, E.H., Du, W. Congruences modulo 7 for broken 3-diamond partitions. Ramanujan J 50, 253–262 (2019). https://doi.org/10.1007/s11139-018-0043-6

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  • DOI: https://doi.org/10.1007/s11139-018-0043-6

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