Bijections between bar-core and self-conjugate core partitions

  • Jane Y. X. YangEmail author


In the recent decades, core partitions and bar core partitions have attracted much attention from combinatorial researchers. In this paper, we build a bijection between \(\bar{t}\)-cores and self-conjugate t-cores for odd integers \(t\ge 3\). To this end, we first establish a bijection between \(\bar{t}\)-cores and self-conjugate \((t-1)\)-cores for odd integers \(t\ge 3\), then we prove that this bijection can be extended to a bijection between self-conjugate \((t-1)\)-cores and self-conjugate t-cores for odd integers \(t\ge 3\).


Bijection Core partition Bar partition Self-conjugate partition Young diagram 

Mathematics Subject Classification

05A17 05A19 



The author appreciates the anonymous referee for his/her helpful comments and language editing which have greatly improved the quality of this manuscript.


  1. 1.
    Andrews, G.E.: Generalized Frobenius Partitions. American Mathematical Society, Providence (1984)CrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  3. 3.
    Baldwin, J., Depweg, M., Ford, B., Kunin, A., Sze, L.: Self-conjugate \(t\)-core partitions, sums of squares, and \(p\)-blocks of \(A_n\). J. Algebra 297, 438–452 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bessenrodt, C.: Bar weights of bar partitions and spin character zeros. J. Algebr. Combin. 26, 107–124 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bessenrodt, C., Olsson, J.B.: Spin block inclusions. J. Algebra 306, 3–16 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Deng, C.: Even \((\bar{s},\bar{t})\)-core partitions and self-associate characters of \(\tilde{S}_n\). arXiv:1508.01462v2
  7. 7.
    Ford, B., Mai, H., Sze, L.: Self-conjugate simultaneous \(p\)- and \(q\)-core partitions and blocks of \(A_n\). J. Number Theory 129, 858–865 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gramain, J.B., Nath, R.: On core and bar-core partitions. Ramanujan J. 27, 229–233 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kiming, I.: On the existence of \(\bar{p}\)-core partitions of natural numbers. Q. J. Math. 48, 59–66 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Clarendon Press, Oxford (1979)zbMATHGoogle Scholar
  11. 11.
    Morris, A.O.: The spin represetations of the symmetric group. Can. J. Math. 17, 543–549 (1965)CrossRefGoogle Scholar
  12. 12.
    Morris, A.O., Yaseen, A.K.: Some combinatorial results involving shifted Young diagrams. Math. Proc. Camb. Philos. Soc. 99, 23–31 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nath, R., Sellers, J.A.: Congruences for spin characters of the double covers of the symmetric and alternating groups. Adv. Appl. Math. 80, 114–130 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Olsson, J.B.: A theorem on the cores of partitions. J. Combin. Theory Ser. A 116, 733–740 (2009)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceChongqing University of Posts and TelecommunicationsChongqingPeople’s Republic of China

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