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    Chen, Shi-Chao 2016. Congruences and asymptotics of Andrews' singular overpartitions. Journal of Number Theory, Vol. 164, p. 343.


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    Wang, Liuquan 2016. Arithmetic identities and congruences for partition triples with 3-cores. International Journal of Number Theory, Vol. 12, Issue. 04, p. 995.


    Wang, Liuquan 2016. Explicit formulas for partition pairs and triples with 3-cores. Journal of Mathematical Analysis and Applications, Vol. 434, Issue. 2, p. 1053.


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    Dai, Haobo and Niu, Chuanze 2015. On the distribution of $$2^t$$ 2 t -core partitions modulo $$2^j$$ 2 j. The Ramanujan Journal, Vol. 38, Issue. 1, p. 115.


    Xia, Ernest X. W. 2015. Arithmetic properties of bipartitions with 3-cores. The Ramanujan Journal, Vol. 38, Issue. 3, p. 529.


    YAO, OLIVIA X. M. 2015. INFINITE FAMILIES OF CONGRUENCES MODULO 3 AND 9 FOR BIPARTITIONS WITH 3-CORES. Bulletin of the Australian Mathematical Society, Vol. 91, Issue. 01, p. 47.


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    Chen, Shi-Chao 2014. Congruences for a Certain Partition Function. Annals of Combinatorics, Vol. 18, Issue. 4, p. 607.


    Lin, Bernard L.S. 2014. Some results on bipartitions with 3-core. Journal of Number Theory, Vol. 139, p. 44.


    LIN, BERNARD L. S. and WANG, ANDREW Y. Z. 2014. GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES. Bulletin of the Australian Mathematical Society, Vol. 90, Issue. 02, p. 204.


    Park, Yoon Kyung 2014. IDENTITIES FOR 3-CORE AND 5-CORE PARTITIONS. Journal of the Korean Mathematical Society, Vol. 51, Issue. 2, p. 225.


    XIA, ERNEST X. W. and YAO, OLIVIA X. M. 2014. A PROOF OF KEITH'S CONJECTURE FOR 9-REGULAR PARTITIONS MODULO 3. International Journal of Number Theory, Vol. 10, Issue. 03, p. 669.


    Yao, Olivia X.M. 2014. New congruences modulo powers of 2 and 3 for 9-regular partitions. Journal of Number Theory, Vol. 142, p. 89.


    Chen, Shi-Chao 2013. Ramanujan-type congruences for certain generating functions. Lithuanian Mathematical Journal, Vol. 53, Issue. 4, p. 381.


    Xia, Ernest X. W. and Yao, Olivia X. M. 2013. Analogues of Ramanujan’s partition identities. The Ramanujan Journal, Vol. 31, Issue. 3, p. 373.


    Yao, Olivia X.M. and Xia, Ernest X.W. 2013. New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions. Journal of Number Theory, Vol. 133, Issue. 6, p. 1932.


    BARUAH, NAYANDEEP DEKA and SARMAH, BIPUL KUMAR 2012. IDENTITIES FOR SELF-CONJUGATE 7- AND 9-CORE PARTITIONS. International Journal of Number Theory, Vol. 08, Issue. 03, p. 653.


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  • Bulletin of the Australian Mathematical Society, Volume 79, Issue 3
  • June 2009, pp. 507-512

ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

  • MICHAEL D. HIRSCHHORN (a1) and JAMES A. SELLERS (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972709000136
  • Published online: 01 June 2009
Abstract
Abstract

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

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Corresponding author
For correspondence; e-mail: sellersj@math.psu.edu
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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]N. Baruah and B. Berndt , ‘Partition identities and Ramanujans modular equations’, J. Combin. Theory Ser. A 114(6) (2007), 10241045.

[2]J. Borwein and P. Borwein , ‘A cubic counterpart of Jacobi’s identity and the AGM’, Trans. Amer. Math. Soc. 323(2) (1991), 691701.

[3]J. Borwein , P. Borwein and F. Garvan , ‘Some cubic modular identities of Ramanujan’, Trans. Amer. Math. Soc. 343(1) (1994), 3547.

[4]A. Granville and K. Ono , ‘Defect zero p-blocks for finite simple groups’, Trans. Amer. Math. Soc. 348(1) (1996), 331347.

[6]M. Hirschhorn , F. Garvan and J. Borwein , ‘Cubic analogues of the Jacobian theta function θ(z,q)’, Canad. J. Math. 45(4) (1993), 673694.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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