hello95
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Hello,
I was wondering if this problem is one that is solvable, and if so, if I would be able to solve it in a reasonable amount of time.
Basically, say you have n blocks. You can arrange this blocks in stacks, so for example, if you had 3 blocks, you could make a stack of 2, and then add a stack of 1. The question is, for these n blocks, how many unique ways are there of arranging them in stacks (the order is irrelevant). I got the idea from a theorem in abstract algebra, which says that a finite abelian group is isomorphic to the direct product of cyclic groups with orders equal to some integral power of a prime. Since such a direct product would be isomorphic to a cyclic group equal to the product of all of the prime power orders, then I could find a way to represent all of the possible forms of a cyclic group of order (2^7)(5^2) by simply taking the number of ways of stacking 7 blocks and multiplying it by the number of ways of stacking 2 blocks.
Another way of looking at the problem would be, if you have n star symbols (*), how many ways would there be of grouping certain numbers of those stars together so that the sum of all of the elements in all of the groupings equaled n:
(*)(*)(*)(*), (**)(**), (*)(***), (****)
Once again, order is irrelevant.
If you know the answer, please do not post it. Ultimately I just want to know if this problem is feasible for someone with a decent knowledge (but my no means expansive) of combinatorics to attempt.
Thank you.
I was wondering if this problem is one that is solvable, and if so, if I would be able to solve it in a reasonable amount of time.
Basically, say you have n blocks. You can arrange this blocks in stacks, so for example, if you had 3 blocks, you could make a stack of 2, and then add a stack of 1. The question is, for these n blocks, how many unique ways are there of arranging them in stacks (the order is irrelevant). I got the idea from a theorem in abstract algebra, which says that a finite abelian group is isomorphic to the direct product of cyclic groups with orders equal to some integral power of a prime. Since such a direct product would be isomorphic to a cyclic group equal to the product of all of the prime power orders, then I could find a way to represent all of the possible forms of a cyclic group of order (2^7)(5^2) by simply taking the number of ways of stacking 7 blocks and multiplying it by the number of ways of stacking 2 blocks.
Another way of looking at the problem would be, if you have n star symbols (*), how many ways would there be of grouping certain numbers of those stars together so that the sum of all of the elements in all of the groupings equaled n:
(*)(*)(*)(*), (**)(**), (*)(***), (****)
Once again, order is irrelevant.
If you know the answer, please do not post it. Ultimately I just want to know if this problem is feasible for someone with a decent knowledge (but my no means expansive) of combinatorics to attempt.
Thank you.