The Ramanujan Journal

, Volume 47, Issue 2, pp 339–381 | Cite as

Partitions and Sylvester waves

  • Cormac O’SullivanEmail author


The restricted partition function \(p_N(n)\) counts the partitions of the integer n into at most N parts. In the nineteenth century, Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as N and n both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to \(p_N(n)\) in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm.


Restricted partitions Sylvester waves Asymptotics Saddle-point method 

Mathematics Subject Classification

11P82 41A60 


  1. 1.
    Apostol, T.M.: On the Lerch zeta function. Pac. J. Math. 1, 161–167 (1951)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beck, M., Gessel, I.M., Komatsu, T.: The polynomial part of a restricted partition function related to the Frobenius problem. Electron. J. Comb. 8(1), Note 7 (2001)Google Scholar
  3. 3.
    Beck, M., Diaz, R., Robins, S.: The Frobenius problem, rational polytopes, and Fourier-Dedekind sums. J. Number Theory 96(1), 1–21 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cayley, A.: Researches on the partition of numbers. Philos. Trans. R. Soc. Lond. 146, 127–140 (1856)CrossRefGoogle Scholar
  5. 5.
    Campbell, J.A., Olof Fröman, P., Walles, E.: Explicit series formulae for the evaluation of integrals by the method of steepest descents. Stud. Appl. Math. 77(2), 151–172 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Comtet, L.: Advanced Combinatorics, enlarged edition. D. Reidel Publishing Co., Dordrecht (1974). The art of finite and infinite expansionsCrossRefGoogle Scholar
  7. 7.
    Drmota, M., Gerhold, S.: Disproof of a conjecture by Rademacher on partial fractions. Proc. Am. Math. Soc. B 1, 121–134 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dowker, J.S.: On Sylvester waves and restricted partitions. arXiv:1302.6172
  9. 9.
    Dowker, J.S.: Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves. arXiv:1108.1760
  10. 10.
    Dilcher, K., Vignat, C.: An explicit form of the polynomial part of a restricted partition function. Res. Number Theory 3(1), 12 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fel, L.G.: Rubinstein, Boris Y.: Sylvester waves in the Coxeter groups. Ramanujan J. 6(3), 307–329 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Glaisher, J.W.L.: Formulae for partitions into given elements, derived from Sylvester’s theorem. Q. J. Pure Appl. Math. 40, 275–348 (1909)zbMATHGoogle Scholar
  13. 13.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford. Revised by D. R. Heath-Brown and J. H, Silverman, With a foreword by Andrew Wiles (2008)Google Scholar
  14. 14.
    Maximon, L.C.: The dilogarithm function for complex argument. Proc. R. Soc. Lond. A 459(2039), 2807–2819 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    O’Sullivan, C.: On the partial fraction decomposition of the restricted partition generating function. Forum Math. 27(2), 735–766 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    O’Sullivan, C.: Asymptotics for the partial fractions of the restricted partition generating function I. Int. J. Number Theory 12(6), 1421–1474 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    O’Sullivan, C.: Asymptotics for the partial fractions of the restricted partition generating function II. Integers 16, A78 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    O’Sullivan, C.: Zeros of the dilogarithm. Math. Comput. 85(302), 2967–2993 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    O’Sullivan, C.: Revisiting the saddle-point method of Perron. arXiv:1702.03611 (2017)
  20. 20.
    Perron, O.: Über die näherungsweise Berechnung von Funktionen großer Zahlen. Sitzungsber. Bayr. Akad. Wissensch. (Münch. Ber.), pp. 191–219 (1917)Google Scholar
  21. 21.
    Rademacher, H.: Topics in Analytic Number Theory. Springer, New York (1973). Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169Google Scholar
  22. 22.
    Rubinstein, B.Y., Fel, L.G.: Restricted partition functions as Bernoulli and Eulerian polynomials of higher order. Ramanujan J. 11(3), 331–347 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sylvester, J.J.: On subvariants, i.e. semi-invariants to binary quantics of an unlimited order: excursus on rational fractions and partitions. Am. J. Math. 5(1), 119–136 (1882)Google Scholar
  24. 24.
    Sills, A.V., Zeilberger, D.: Formulæ for the number of partitions of \(n\) into at most \(m\) parts (using the quasi-polynomial ansatz). Adv. Appl. Math. 48(5), 640–645 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Szekeres, G.: An asymptotic formula in the theory of partitions. Q. J. Math. Oxf. 2(2), 85–108 (1951)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zagier, D.: The Dilogarithm Function. Frontiers in Number Theory, Physics, and Geometry. II, pp. 3–65. Springer, Berlin (2007)Google Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Cuny Graduate CenterNew YorkUSA

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