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Using natural numbers 1,2,...n, in how many ways can the number n be formed from the sum of one or more smaller natural numbers? I thought it would be an easy problem but i couldn't figure it out. Example: For n = 4, we have 4,1 + 3, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1 for a total of 5.

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What prevents you from writing $4=1+1+1+1$? – Peter Košinár Jun 7 '13 at 14:38
Thanks, i corrected that. I guess i can also add 4 if i changed the question to 1 or more sums. – danny Jun 7 '13 at 14:40
up vote 4 down vote accepted

The answer is complicated and you've missed $4=1+1+1+1$ in your question. You are looking for partitions - the link has a good bibliography. The exact formula is surprisingly complicated. Your question involves subtracting 1 from the number of partitions as conventionally calculated.

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Thank you very much. Yes that is what I wanted to do. Indeed it seems complicated than i thought. – danny Jun 7 '13 at 14:46
The problem gets a lot easier if one treats different orders of the same numbers as different (e.g. $1+3$ being different from $3+1$)... but that's a completely different beast to tackle :-) – Peter Košinár Jun 7 '13 at 14:47
Luckily for my problem permutations don't matter. – danny Jun 7 '13 at 14:49

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