Advertisement

Research in Number Theory

, 5:25 | Cite as

Unimodal sequence generating functions arising from partition ranks

  • Kathrin BringmannEmail author
  • Chris Jennings-Shaffer
Research
  • 10 Downloads

Abstract

In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a relation between the rank of strongly unimodal sequences and the rank of integer partitions.

Keywords

Unimodal sequences Strongly unimodal sequences Partitions Overpartitions Unimodal ranks Partition ranks Dyson rank \(M_2\)-rank Asymptotics Modular forms Mock modular forms 

Mathematics Subject Classification

05A16 11F03 11P81 11P82 

Notes

Acknowledgements

We thank Amanda Folsom and Jeremy Lovejoy for bringing [11] to our attention. We also thank the anonymous referees for their helpful comments and pointing out various typos in an earlier version of this manuscript.

References

  1. 1.
    Andrews, G.: The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, vol. 2. Addison-Wesley Publishing Co., Reading (1976)Google Scholar
  2. 2.
    Andrews, G.: Ramanujan’s “lost” notebook. IV. Stacks and alternating parity in partitions. Adv. Math. 53, 55–74 (1984)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrews, G.: $q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, No. 66 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical (1986)Google Scholar
  4. 4.
    Andrews, G., Berndt, B.: Ramanujan’s Lost Notebook. Part I. Springer, New York (2005)zbMATHGoogle Scholar
  5. 5.
    Andrews, G., Berndt, B.: Ramanujan’s Lost Notebook. Part II. Springer, New York (2009)zbMATHGoogle Scholar
  6. 6.
    Andrews, G., Warnaar, S.: The Bailey transform and false theta functions. Ramanujan J. 14, 173–188 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Andrews, G., Dyson, F., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91, 391–407 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Atkin, A., Swinnerton-Dyer, P.: Some properties of partitions. Proc. Lond. Math. Soc. 4, 84–106 (1954)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Auluck, F.: On some new types of partitions associated with generalized Ferrers graphs. Proc. Camb. Philos. Soc. 47, 84–106 (1951)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Baker, A.: A Comprehensive Course in Number Theory. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  11. 11.
    Barnett, M., Folsom, A., Wesley, W.: Rank generating functions for odd-balanced unimodal sequences, quantum Jacobi forms, and mock Jacobi forms, preprint (2018)Google Scholar
  12. 12.
    Barnett, M., Folsom, A., Ukogu, O., Wesley, W., Xu, H.: Quantum Jacobi forms and balanced unimodal sequences. J. Number Theory 186, 16–34 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Berkovich, A., Garvan, F.: Some observations on Dyson’s new symmetries of partitions. J. Comb. Theory Ser. A 100, 61–93 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bringmann, K., Jennings-Shaffer, C., Mahlburg, K.: On a Tauberian theorem of Ingham and Euler-Maclaurin summation, in preparationGoogle Scholar
  15. 15.
    Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Ann. Math. 171, 419–449 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bringmann, K., Creutzig, T., Rolen, L.: Negative index Jacobi forms and quantum modular forms. Res. Math. Sci. 1, 11 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications, vol. 64. American Mathematical Society, Providence (2017)CrossRefGoogle Scholar
  18. 18.
    Bryson, J., Ono, K., Pitman, S., Rhoades, R.: Unimodal sequences and quantum and mock modular forms. Proc. Natl. Acad. Sci. USA 109, 16063–16067 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dyson, F.: Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)MathSciNetGoogle Scholar
  20. 20.
    Garvan, F.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11. Trans. Am. Math. Soc. 305, 47–77 (1988)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  22. 22.
    Hardy, G., Ramanujan, S.: Asymptotic formulaae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kim, B., Lim, S., Lovejoy, J.: Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity. Proc. Am. Math. Soc. 144, 3687–3700 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lovejoy, J.: Lacunary partition functions. Math. Res. Lett. 9, 191–198 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lovejoy, J.: Overpartitions and real quadratic fields. J. Number Theory 106, 178–186 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lovejoy, J.: Rank and conjugation for the Frobenius representation of an overpartition. Ann. Comb. 9, 321–334 (2005)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lovejoy, J.: Rank and conjugation for a second Frobenius representation of an overpartition. Ann. Comb. 12, 101–113 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lovejoy, J., Osburn, R.: $M_2$-rank differences for partitions without repeated odd parts. J. Theor. Nombres Bordeaux 21, 313–334 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Marcus, D.A.: Number Fields, 2nd edn. Springer, Cham (2018)CrossRefGoogle Scholar
  30. 30.
    Mortenson, E.: On the dual nature of partial theta functions and Appell-Lerch sums. Adv. Math. 264, 236–260 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ramanujan, S.: Congruence properties of partitions. Math. Z. 9, 147–153 (1921)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rhoades, R.: Asymptotics for the number of strongly unimodal sequences. Int. Math. Res. Not. 2014(3), 700–719 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Stanley, R.: Unimodal sequences arising from Lie algebras. Combinatorics, representation theory and statistical methods in groups, Lecture Notes in Pure and Appl. Math. 57, 127–136, Dekker, New York (1980)Google Scholar
  34. 34.
    Stanley, R.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986). Ann. N. Y. Acad. Sci. 576, 500–535 (1989)CrossRefGoogle Scholar
  35. 35.
    Wright, E.: Stacks. Q. J. Math. Oxford Ser. 19, 313–320 (1968)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zagier, D.: Quantum modular forms. Quanta Maths 11, 659–675 (2010)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zwegers, S.: Mock theta functions, Ph.D. Thesis, Universiteit Utrecht (2002)Google Scholar
  38. 38.
    Zwegers, S.: Multivariable Appell functions and nonholomorphic Jacobi forms. Res. Math. Sci. 6, 16 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical and Natural Sciences, Mathematical InstituteUniversity of CologneCologneGermany

Personalised recommendations