Let n be a natural number, then s is the number of ways to write n as the sum of at least one natural number. What is the number s called?

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- Jun 22nd 2010, 03:30 PMChris11What are the following types of numbers called?
Let n be a natural number, then s is the number of ways to write n as the sum of at least one natural number. What is the number s called?

- Jun 22nd 2010, 03:36 PMundefined
If the addends are not necessarily distinct and order is not considered, then we have Partition Function P. And here is the Wikipedia article.

- Jun 22nd 2010, 03:39 PMChris11
What about when order is considered?

- Jun 22nd 2010, 03:41 PMundefined
This is the number of compositions of n.

- Jun 22nd 2010, 03:45 PMChris11
Thanks for replying. I made the conjecture that the number of compositions is $\displaystyle 2^{n-1}$, but I haven't been able to prove it. I don't understand the proof on wikipedia much either... I shamefully don't know anything about combinatorical proof..

- Jun 22nd 2010, 04:03 PMundefined
- Jun 22nd 2010, 06:33 PMChris11
yeah, it makes alot more sense. Thanks.