The Ramanujan Journal

, Volume 38, Issue 1, pp 61–73 | Cite as

Log-concavity of the partition function

  • Stephen DeSalvo
  • Igor Pak


We prove that the partition function \(p(n)\) is log-concave for all \(n>25\). We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based on Lehmer’s estimates on the remainders of the Hardy–Ramanujan and the Rademacher series for \(p(n)\).


Integer partition Partition function Log-concave sequence Asymptotic analysis Error estimates 

Mathematics Subject Classification

05A17 11N37 65G99 



The authors would like to thank William Chen, Christian Krattenthaler, Karl Mahlburg, Bruce Rothschild, Bruce Sagan for helpful remarks and suggestions. The authors are grateful to Richard Arratia for introducing us to the problem, to David Moews for bringing the work of Lehmer to our attention, and to Janine Janoski for informing us about the status of [14].


  1. 1.
    Andrews, G.E.: Combinatorial proof of a partition function limit. Am. Math. Mon. 78, 276–278 (1971)MATHCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  3. 3.
    Bessenrodt, C., Ono, K.: Maximal multiplicative properties of partitions.
  4. 4.
    Bóna, M.: A combinatorial proof of the log-concavity of a famous sequence counting permutations. Electron. J. Comb. 11(2), 4 (2004)Google Scholar
  5. 5.
    Brenti, F.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. Contemp. Math. 178, 71–89 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bringmann, K., Mahlburg, K.: Asymptotic formulas for stacks and unimodal sequences. J. Comb. Theory Ser. A 126, 194–215 (2014)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, W.Y.C.: Recent developments on log-concavity and q-log-concavity of combinatorial polynomials. In: FPSAC 2010 Conference Talk Slides. (2010)
  8. 8.
    Chen, W.Y.C., Wang, L.X.W., Xie, G.Y.B.: Finite differences of the logarithm of the partition function. arXiv:1407.0177 (2014)
  9. 9.
    Erdős, P.: On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. 43, 437–450 (1942)CrossRefGoogle Scholar
  10. 10.
    Gupta, H.: Tables of Partitions. Indian Mathematical Society, Madras (1939)Google Scholar
  11. 11.
    Han, H., Seo, S.: Combinatorial proofs of inverse relations and log-concavity for Bessel numbers. Eur. J. Comb. 29, 1544–1554 (2008)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hardy, G.H.: Twelve lectures on subjects suggested by his life and work. Cambridge University Press, Cambridge (1940)Google Scholar
  13. 13.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Janoski, J.E.: A Collection of problems in combinatorics. Ph.D. thesis, Clemson University. (2012)
  15. 15.
    Krattenthaler, C.: Combinatorial proof of the log-concavity of the sequence of matching numbers. J. Comb. Theory Ser. A 74, 351–354 (1996)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lehmer, D.H.: On the Hardy–Ramanujan series for the partition function. J. Lond. Math. Soc. 12, 171–176 (1937)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lehmer, D.H.: On the series for the partition function. Trans. AMS 43, 271–295 (1938)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lehmer, D.H.: On the remainders and convergence of the series for the partition function. Trans. AMS 46, 362–373 (1939)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liu, L.L., Wang, L.L.: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39, 453–476 (2007)MATHCrossRefGoogle Scholar
  20. 20.
    Rademacher, H.: A convergent series for the partition function p(n). Proc. Natl. Acad. Sci. USA 23, 78–84 (1937)CrossRefGoogle Scholar
  21. 21.
    Sagan, B.E.: Inductive and injective proofs of log concavity results. Discret. Math. 68, 281–292 (1988)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Stanley, R.P.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Ann. New York Acad. Sci. 576, 500–535 (1989)CrossRefGoogle Scholar
  23. 23.
    Sun, Z.-W.: Conjectures involving arithmetical sequences, in Number theory–arithmetic in Shangri-La. Hackensack, World Scientific (2013)Google Scholar
  24. 24.
    Sun, Z.-W.: On a sequence involving sums of primes. Bull. Aust. Math. Soc. 88, 197–205 (2013)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Wright, E.: Stacks. III. Q. J. Math. Ser. 23, 153–158 (1972)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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