Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T14:22:18.794Z Has data issue: false hasContentIssue false
  • English
  • Français

Computations of the partition function

Published online by Cambridge University Press:  01 August 2016

P. Shiu
P. Shiu*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU
Get access Rights & Permissions [Opens in a new window]

Extract

How many ways are there of making up £1 using coins? Omitting the use of the £1 coin, we are asking for the number of ways of ‘partitioning’ 100 using parts of 1, 2, 5, 10, 20, 50. For n ≥ 0, let f (n) denote the number of solutions to the equation

in non-negative integers xi, so that our required number is f (100). It is not difficult to see that f(n) is the coefficient of tn in the product Π(1 + tk + t2k +… ) over the parts k = 1, 2, 5, 10, 20, 50.

Type
Articles
Information
The Mathematical Gazette , Volume 81 , Issue 490 , March 1997 , pp. 45 - 52
Copyright
Copyright © The Mathematical Association 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Shiu, CM. and Shiu, P., Stamps and coins: two partition problems, Mathematical Spectrum, 13, (1981) pp. 49–55.Google Scholar
2. Niven, I., Zukerman, H. S. and Montgomery, H. L., An introduction to the theory of numbers, 5th Edition, John Wiley (1991).Google Scholar
3. Rademacher, H., Topics in analytic number theory, Springer-Verlag (1973).Google Scholar
4. Burr, S. A. (Editor), The unreasonable effectiveness of number theory, American Mathematical Society (1992).Google Scholar