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Computing the integer partition function


Authors: Neil Calkin, Jimena Davis, Kevin James, Elizabeth Perez and Charles Swannack
Journal: Math. Comp. 76 (2007), 1619-1638
MSC (2000): Primary 05A17; Secondary 11P81, 11P83
DOI: https://doi.org/10.1090/S0025-5718-07-01966-7
Published electronically: February 28, 2007
MathSciNet review: 2299791
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Abstract: In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of $ p(N)$ for $ N \le 10^9$ for primes up to $ 103$ and small powers of $ 2$ and $ 3$.


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  • 1. S. Ahlgren, Distribution of parity of the partition function in arithmetic progressions, Indag. Math. 10 (1999) 173-181. MR 1816213 (2002i:11102)
  • 2. S. Ahlgren, The partition function modulo composite integers $ M$, Math. Ann. 318 (2000), 795-803. MR 1802511 (2001j:11099)
  • 3. S. Ahlgren and M. Boylan Arithmetic properties of the partition function, Invent. Math. 153 (2003), no. 3, 487-502. MR 2000466 (2004e:11115)
  • 4. S. Ahlgren and M. Boylan Coefficients of half-integral weight modular forms modulo $ l\sp j$, Math. Ann. 331 (2005), no. 1, 219-239. MR 2107445 (2005k:11091)
  • 5. S. Ahlgren and M. Boylan Addendum: "Coefficients of half-integral weight modular forms modulo $ l\sp j$, Math. Ann. 331 (2005), no. 1, 219-239 MR 2107445 (2005k:11091)
  • 6. S. Ahlgren and K. Ono, Congruences and conjectures for the partition function, Contemporary Math., 291 (2001) 1-10. MR 1874518 (2002j:11120)
  • 7. S. Ahlgren and K. Ono, Congruence properties for the partition function, Proceedings of the National Academy of Sciences, 98 no. 9 (2001), 978-984. MR 1862931 (2002k:11187)
  • 8. G. E. Andrews, (1998) The theory of partitions, (1998) Cambridge University Press, Cambridge. MR 1634067 (99c:11126)
  • 9. A. O. L. Atkin, Multiplicative congruence properties and density problems for $ p(n)$, Proc. London Math. Soc., 18 (1968) 563-576. MR 0227105 (37:2690)
  • 10. A. O. L. Atkin and J. N. O'Brien, Some properties of $ p(n)$ and $ c(n)$ modulo powers of 13, Trans. Amer. Math. Soc., 126 (1967) 442-459. MR 0214540 (35:5390)
  • 11. A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics, Proc. Sympos. Pure Math., UCLA, 1968, 19 (1971) 1-25. MR 0337781 (49:2550)
  • 12. J. Bruinier and K. Ono, Fourier coefficients of half-integral weight modular forms, J. Number Th. 99 (2003), 164-179. MR 1957250 (2004b:11056)
  • 13. J. Bruinier and K. Ono, Fourier coefficients of half-integral weight modular forms (corrigendum), J. Number Th. 104 (2004), 378-379. MR 2029514 (2004j:11044)
  • 14. Heath-Brown, D. R. Zero-free regions for Dirichlet $ L$-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. (3) 64 (1992), no. 2, 265-338. MR 1143227 (93a:11075)
  • 15. O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 (1959), 377-378. MR 0117213 (22:7995)
  • 16. T. Klove, Recurrence formulae for the coefficients of modular forms and congruences for the partition function and for the coefficients of $ j(\tau)$, $ (j(\tau)-1728)^{1/2}$, and $ j(\tau)^{1/3}$, Math. Scand. 23 (1969), 133-159. MR 0252324 (40:5545)
  • 17. M. Newman, Periodicity modulo $ m$ and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960) 225-236. MR 0115981 (22:6778)
  • 18. J. -L. Nicolas, I. Z. Ruzsa and A. Sárközy (with an appendix by J. -P. Serre), On the parity of additive representation functions, J. number Theory, 73, (1998) 292-317. MR 1657968 (2000a:11151)
  • 19. K. Ono, Distribution of the partition function modulo $ m$, Ann. Math. 151 (2000) 293-307. MR 1745012 (2000k:11115)
  • 20. K. Ono, The partition function in arithmetic progressions, Math. Ann. 312 (1998) 251-260. MR 1671788 (2000j:11153)
  • 21. T. R. Parkin and D. Shanks, (1967) On the distribution of parity in the partition function, Math. Comp., 21, 466-480. MR 0227126 (37:2711)
  • 22. H. Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. MR 0364103 (51:358)
  • 23. S. Ramunajan, Congruence properties of partitions, Proceedings of the London Mathematical Society, 19 (1919) 207-210.
  • 24. R. L. Weaver, New congruences for the partition function, Ramanujan J., 5 (2001), no. 1, 53-63. MR 1829808 (2002h:11102)
  • 25. N. Calkin, J. Davis, K. James, E. Perez, and C. Swannack, cited 2006: Statistics of the Integer Partition Function. Available online at http://allegro.mit.edu/~swannack/ Projects/partitionStats/index.html

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Additional Information

Neil Calkin
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
Email: calkin@clemson.edu

Jimena Davis
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
Email: jldavis9@unity.ncsu.edu

Kevin James
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
Email: kevja@clemson.edu

Elizabeth Perez
Affiliation: Applied Mathematics and Statistics, The Johns Hopkins University, G.W.C. Whiting School of Engineering, 302 Whitehead Hall, 3400 North Charles Street, Baltimore, Maryland 21218-2682
Email: eaperez@ams.jhu.edu

Charles Swannack
Affiliation: Department of Electrical and Computer Engineering, Clemson University, Clemson, South Carolina 29634
Address at time of publication: Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: swannack@mit.edu

DOI: https://doi.org/10.1090/S0025-5718-07-01966-7
Keywords: Partition function, discrete fast Fourier transforms
Received by editor(s): March 11, 2005
Received by editor(s) in revised form: July 10, 2006
Published electronically: February 28, 2007
Additional Notes: The authors were partially supported by NSF grant DMS-0139569
The third author was partially supported by NSF grant DMS-0090117
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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