Skip to main content
Springer Nature Link
Log in

Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Atkin and Garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks. Here we consider a combinatorial interpretation of these moments. This requires the introduction of a new representation for partitions, the Durfee symbol, and subsequent refinements. This in turn leads us to a variety of new congruences for our ‘marked’ Durfee symbols much in the spirit of Dyson’s original conjectures on the ranks of partitions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Agarwal, A.K., Andrews, G.E.: Rogers–Ramanujan identities for partitions with “N copies of N”. J. Comb. Theory, Ser. A, 45, 40–49 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ahlgren, S., Ono, K.: Addition and counting: The arithmetic of partitions. Notices Am. Math. Soc. 48, 978–984 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Alladi, K.: The method of weighted words and applications to partitions. In: David, S. (ed.) Number Theory, pp. 1–36. Cambridge University Press, Cambridge (1995)

  4. Alladi, K., Andrews, G.E., Berkovich, A.: A new four parameter q-series identity and its partition implications. Invent. Math. 153, 231–260 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Andrews, G.E.: Problems and prospects for basic hypergeometric functions. In: Askey, R. (ed.) Theory and Appl. of Special Functions, pp. 191–224. Academic Press, New York (1975)

  6. Andrews, G.E.: The theory of partitions. In: Rota, G.-C. (ed.) Encycl. of Math. and Its Appl. Addison-Wesley, Reading, (1976) (Reissued: Cambridge University Press, New York, 1998)

  7. Andrews, G.E., Crippa, D., Simon, K.: q-Series arising from the study of random graphs. SIAM J. Discrete Math. 10, 41–56 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Andrews, G.E., Freitas, P.: Extension of Abel’s Lemma with q-series implications. Ramanujan J. 10, 137–152 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Andrews, G.E., Garvan, F.: Dyson’s crank of a partition. Bull. Am. Math. Soc. 18, 167–171 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Atkin, A.O.L., Garvan, F.: Relations between the ranks and cranks of partitions. Ramanujan J. 7, 343–366 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Atkin A.O.L., Swinnerton-Dyer, H.P.F.: Some properties of partitions. Proc. Lond. Math. Soc., III Ser. 4, 84–106 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bringmann, K.: Asymptotics for rank partition functions. (to appear)

  13. Bringmann, K., Ono, K.: The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165, 243–266 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Submitted

  15. Dilcher, K.: Some q-series identities related to divisor functions. Discrete Math. 145, 98–110 (1995)

    Article  MathSciNet  Google Scholar 

  16. Dyson, F.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944) (Reprinted: Selected Papers, Am. Math. Soc., Providence, pp. 51–56 (1996))

  17. Fine, N.J.: Basic Hypergeometric Series and Applications. Am. Math. Soc., Providence (1988)

    MATH  Google Scholar 

  18. Garthwaite, S.: The coefficients of the ω(q) mock theta function. Proc. Am. Math. Soc. (to appear)

  19. Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences. Trans. Am. Math. Soc. 305, 47–77 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Garvan, F.G.: Generalizations of Dyson’s rank and non-Rogers–Ramanujan partitions. Manuscr. Math. 84, 343–359 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  22. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)

    MATH  Google Scholar 

  23. Kluyver, J.C.: Vraagstuk XXXVII. (Solution by van Veen, S.C.). Wiskundige Opgaven, 92–93 (1919)

  24. MacMahon, P.A.: The theory of modular partitions. Proc. Camb. Phil. Soc. 21, 197–204 (1923) (Reprinted: Coll. Papers, vol. 1, M.I.T. Press, Cambridge, pp. 1090–1097 (1978))

  25. Mahlburg, K.: Partition congruences and the Andrews–Garvan–Dyson crank. Proc. Nat. Acad. Sci. 102, 15373–15376 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ono, K.: Distribution of the partition function modulo m. Ann. Math. 151, 293–307 (2000)

    Article  MATH  Google Scholar 

  27. Ono, K.: Arithmetic of the partition function. In: Bustoz, J., Suslov, S. (eds.) Proc. NATO Adv. Study Inst. on Special Functions, Special Functions, pp. 243–253. Kluwer, New York (2000)

  28. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Intro. by G.E. Andrews. Narosa, New Delhi (1997)

    Google Scholar 

  29. Sylvester, J.J.: Note on the paper of Mr.Durfee. Johns Hopkins Univ. Circulars 2, pp. 23, 24; 42, 43 (1883) (Reprinted: Coll. Papers, vol. 3, pp. 660–662, Chelsea, New York (1973))

  30. Sylvester, J.J.: A constructive theory of partitions .... Am. J. Math. 5, 251–330 (1882), 6, 334–336 (1884) (Reprinted: Coll. Papers, vol. 4, pp. 1–83, Chelsea, New York (1973))

  31. Uchimura, K.: An identity for the divisor generating function arising from sorting theory. J. Comb. Theory, Ser. A 31, 131–135 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  32. Watson, G.N.: The final problem. J. Lond. Math. Soc. 11, 55–80 (1936)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pennsylvania State University, University Park, PA, 16802, USA

    George E. Andrews

Authors
  1. George E. Andrews
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to George E. Andrews.

Additional information

Mathematics Subject Classification (2000)

05A17, 05A19, 11P83

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andrews, G. Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks. Invent. math. 169, 37–73 (2007). https://doi.org/10.1007/s00222-007-0043-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-007-0043-4

Keywords