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Annals of Combinatorics

, Volume 23, Issue 2, pp 241–248 | Cite as

Partitions with Parts Separated by Parity

  • George E. AndrewsEmail author
Article
  • 38 Downloads

Abstract

There have been a number of papers on partitions in which the parity of parts plays a central role. In this paper, the parts of partitions are separated by parity, either all odd parts are smaller than all even parts or vice versa. This concept first arose in a study related to the third-order mock theta function \(\nu (q)\). The current study also leads back to one of the Ramanujan’s more mysterious functions.

Keywords

Partitions Parity of parts Ramanujan 

Mathematics Subject Classification

11P84 11P83 11P81 

Notes

Acknowledgements

The author wishes to thank Kathy Ji, whose care in reading this paper and comments added greatly to the value.

References

  1. 1.
    Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam (1976) (Reissued: Cambridge University Press, Cambridge (1998))Google Scholar
  2. 2.
    Andrews, G.E.: Unsolved problems: questions and conjectures in partition theory. Amer. Math. Monthly 93(9), 708–711 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews, G.E.: Ramanujan’s “lost” notebook. V. Euler’s partition identity. Adv. in Math. 61(2), 156–164 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andrews, G.E.: Parity in partition identities. Ramanujan J. 23(1-3), 45–90 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Andrews, G.E.: Differences of partition functions: the anti-telescoping method. In: Farkas, H.M., Gunning, R.C., Knopp, M.I., Taylor, B.A. (eds.) From Fourier Analysis and Number Theory to Radon Transformations and Geometry, Dev. Math., Vol. 28, pp. 1–20. Springer, New York (2013)Google Scholar
  6. 6.
    Andrews, G.E.: Integer partitions with even parts below odd parts and mock theta functions. Ann. Comb. 22(3), 433–445 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part I. Springer, New York (2005)zbMATHGoogle Scholar
  8. 8.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part II. Springer, New York (2009)zbMATHGoogle Scholar
  9. 9.
    Andrews, G.E., Dyson, F.J., Hickson, D.R.: Partitions and indefinite quadratic forms. Invent. Math. 91(3), 391–407(1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohen, H.: \(q\)-identities for Maass waveforms. Invent. Math. 91(3), 409–422 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fine, N.J.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence, RI (1988)CrossRefzbMATHGoogle Scholar
  12. 12.
    Franklin, F.: Sur le dévelopment du produit infini \((1-x)(1-x^2)(1-x^3)\ldots \). C. R. Acad. Sci. Paris 82, 448–450 (1881)zbMATHGoogle Scholar
  13. 13.
    Kurşungöz, K.: Parity considerations in Andrews-Gordon identities. European J. Combin. 31(3), 976–1000 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Legendre, A.M.: Théore des Nombres. Fŕmin Didot Frères, Paris (1830)Google Scholar
  15. 15.
    Li, Y., Ngo, H.T., Rhoades, R.C.: Renormalization and quantum modular forms, part I: Maass wave forms. (to appear)Google Scholar
  16. 16.
    Li, Y., Ngo, H.T., Rhoades, R.C.: Renormalization and quantum forms, part II: Mock theta functions. (to appear)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The Pennsylvania State UniversityUniversity ParkUSA

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