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HandWiki. Partition (Number Theory). Encyclopedia. Available online: https://encyclopedia.pub/entry/33028 (accessed on 22 September 2026).
HandWiki. Partition (Number Theory). Encyclopedia. Available at: https://encyclopedia.pub/entry/33028. Accessed September 22, 2026.
HandWiki. "Partition (Number Theory)" Encyclopedia, https://encyclopedia.pub/entry/33028 (accessed September 22, 2026).
HandWiki. (2022, November 04). Partition (Number Theory). In Encyclopedia. https://encyclopedia.pub/entry/33028
HandWiki. "Partition (Number Theory)." Encyclopedia. Web. 04 November, 2022.
Partition (Number Theory)
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In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways:

4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1

The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1. A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.

group representation symmetric group number theory

References

  1. Andrews 1976, p. 199.
  2. Josuat-Vergès, Matthieu (2010), "Bijections between pattern-avoiding fillings of Young diagrams", Journal of Combinatorial Theory, Series A 117 (8): 1218–1230, doi:10.1016/j.jcta.2010.03.006 . https://dx.doi.org/10.1016%2Fj.jcta.2010.03.006
  3. Andrews 1976, p. 69.
  4. Hardy & Wright 2008, p. 380.
  5. Alder, Henry L. (1969). "Partition identities - from Euler to the present". American Mathematical Monthly 76 (7): 733–746. doi:10.2307/2317861. http://www.maa.org/programs/maa-awards/writing-awards/partition-identities-from-euler-to-the-present. 
  6. Hardy & Wright 2008, p. 362.
  7. Hardy & Wright 2008, p. 368.
  8. Hardy & Wright 2008, p. 365.
  9. Notation follows Abramowitz & Stegun 1964, p. 825
  10. Andrews, George E. (1971). Number Theory. Philadelphia: W. B. Saunders Company. pp. 149–50. 
  11. Abramowitz & Stegun 1964, p. 825, 24.2.2 eq. I(B)
  12. Abramowitz & Stegun 1964, p. 826, 24.2.2 eq. II(A)
  13. Richard Stanley, Enumerative Combinatorics, volume 1, second edition. Cambridge University Press, 2012. Chapter 1, section 1.7.
  14. Hardy, G.H. (1920). Some Famous Problems of the Theory of Numbers. Clarendon Press. https://archive.org/details/in.ernet.dli.2015.84630. 
  15. Andrews 1976, pp. 70,97.
  16. Nathanson 2000, pp. 475-85.
  17. Erdős, Pál (1942). "On an elementary proof of some asymptotic formulas in the theory of partitions". Ann. Math.. (2) 43 (3): 437–450. doi:10.2307/1968802.  https://dx.doi.org/10.2307%2F1968802
  18. Nathanson 2000, p. 495.
  19. Nathanson 2000, pp. 458-64.
  20. Andrews 1976, pp. 33–34.
  21. see, e.g., Stanley 1999, p. 58
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