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Before some time, I came upon this exercise :

For every value of $n \in \mathbb N$, find all the partitions of the set : $\{1,2,\dots , n\}$, without using the Bell Number

I know that you can find a number for the partitions by working out a relation from, let's say, the group $\{1,2,3\}$ and $\{1,2,\dots , n+1\}$ with the usage of the Bell Number, which leads us to :

$$B_n = \sum_{k = 0}^n \binom{n}{k} B_{k}$$

My question though is how can we answer on this question, without using the Bell Number ? I'm not supposed to know that the Bell Number is or represents, so I would really like if someone could provide a detailed answer about how I can find or calculate the number of all the partitions of the given set without using the Bell Number.

Peter Taylor
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Rebellos
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    Here is with/without Bell numbers:https://math.stackexchange.com/questions/289016/partitions-and-bell-numbers. – Dietrich Burde Nov 07 '17 at 20:06
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    I don't really know if this is what you're looking for, but maybe it is somewhat helpful. The Stirling number of the second kind, $n\brace k$, counts the number of partitions of ${1,2,\ldots, n}$ into exactly $k$ blocks. So the total number of partitions of this set is $$\sum_{k=0}^n {n\brace k}$$ – Dave Nov 07 '17 at 20:07
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    What is the question asking for? Is it asking for a closed form expression for the number of partitions? Is it asking for a recursive formula for the number of partitions? Is it asking for an algorithm to print out all the partitions? What you wrote was "find all the partitions". If you meant, "find the number of partitions", then please clarify that. – Zach Teitler Nov 07 '17 at 20:50
  • @ZachTeitler The question says: "find all the partitions". I guess it means the number of partitions ? How would it be possible to find all the partitions, since they're infinite ? – Rebellos Nov 07 '17 at 20:52
  • For a given $n$ there are only a finite number of partitions, so it would be possible to find them all. If you want to find all the partitions for every $n,$ that is an infinite task. Likewise, if you want to find the number of partitions for every $n,$ that is an infinite task, since it's an infinite set of numbers. – bof Nov 08 '17 at 01:09
  • I'm not sure what "without using the Bell number" means. If the desired answer is the number of partitions, that number is by definition the Bell number. So the question is asking us to "find the Bell number without using the Bell number"? It seems to my that any method of finding the Bell number would necessarily be "without using the Bell number"; how could you use the Bell number before you found it? – bof Nov 08 '17 at 01:13
  • By the way, what is the author and title of the work you quoted the problem from? It is wrong to quote somebody else's work without acknowledging them. – bof Nov 08 '17 at 01:14
  • @bof I think OP should introduce and define "Carillon Numbers" and use them to solve the problem... :-) – Zach Teitler Nov 08 '17 at 04:06

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