IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4614-3810-6_2.html
   My bibliography  Save this book chapter

Ranks and Cranks, Part I

In: Ramanujan's Lost Notebook

Author

Listed:
  • George E. Andrews

    (The Pennsylvania State University, Department of Mathematics)

  • Bruce C. Berndt

    (University of Illinois at Urbana–Champaign, Department of Mathematics)

Abstract

This chapter is devoted to some of Ramanujan’s most prescient work. A romantic adventure in the theory of partitions began in 1944 when Freeman Dyson defined the rank of a partition to provide a combinatorial interpretation for the famous Ramanujan congruences with moduli 5 and 7. Dyson also conjectured the existence of a second partition statistic which he playfully named the “crank” and hoped it would explain the Ramanujan congruence for the modulus 11. In the early 1950’s, Atkin and Swinnerton-Dyer proved all of Dyson’s conjectures for the rank. Amazingly, all their results are equivalent to two of the entries from the Lost Notebook presented in this chapter. We follow closely the work of Frank Garvan, who first realized all of the above and who laid the groundwork for the discovery of the crank (which was finally presented by Andrews and Garvan in 1988).

Suggested Citation

  • George E. Andrews & Bruce C. Berndt, 2012. "Ranks and Cranks, Part I," Springer Books, in: Ramanujan's Lost Notebook, edition 127, chapter 2, pages 9-44, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-3810-6_2
    DOI: 10.1007/978-1-4614-3810-6_2
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-1-4614-3810-6_2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.