# Thread: Ways to partition [n] (the power set of n)

1. ## Ways to partition [n] (the power set of n)

How many partitions of [n] into two blocks are there? How many partitions of [n] into n-1 blocks are there?

There are $\sum_{k=1}^{floor(\frac{n}{2})}\left({n\atop k}\right)$ partitions of [n] into two blocks (floor(n/2) is the greatest integer less than or equal to (n/2)).

There are $\left({n\atop n-2}\right)$ partitions of [n] into n-1 blocks.

Did I get this one correct?

2. Originally Posted by oldguynewstudent
How many partitions of [n] into two blocks are there? How many partitions of [n] into n-1 blocks are there?

There are $\sum_{k=1}^{floor(\frac{n}{2})}\left({n\atop k}\right)$ partitions of [n] into two blocks (floor(n/2) is the greatest integer less than or equal to (n/2)).

There are $\left({n\atop n-2}\right)$ partitions of [n] into n-1 blocks.

Did I get this one correct?
Stirling numbers of the second kind - Wikipedia, the free encyclopedia

(I've actually done research on this topic!)

3. Originally Posted by oldguynewstudent
How many partitions of [n] into two blocks are there? How many partitions of [n] into n-1 blocks are there?

There are $\sum_{k=1}^{floor(\frac{n}{2})}\left({n\atop k}\right)$ partitions of [n] into two blocks (floor(n/2) is the greatest integer less than or equal to (n/2)).

There are $\left({n\atop n-2}\right)$ partitions of [n] into n-1 blocks.

Did I get this one correct?
What do you mean by "power set of n"? An example of a power set is:

$A=\{1,a\}$

$\mathcal{P}(A)=\{\emptyset,\{1\},\{a\},\{1,a\}\}$

I've noticed in the past you used [n] to mean $\{1,2,\dots,n\}$. So, what are we talking about here?

4. Originally Posted by undefined
What do you mean by "power set of n"? An example of a power set is:

$A=\{1,a\}$

$\mathcal{P}(A)=\{\emptyset,\{1\},\{a\},\{1,a\}\}$

I've noticed in the past you used [n] to mean $\{1,2,\dots,n\}$. So, what are we talking about here?
OOOPS, I was tired. Yes I meant the second. $\{1,2,\dots,n\}$.

5. Originally Posted by oldguynewstudent
How many partitions of [n] into two blocks are there? How many partitions of [n] into n-1 blocks are there?

There are $\sum_{k=1}^{floor(\frac{n}{2})}\left({n\atop k}\right)$ partitions of [n] into two blocks (floor(n/2) is the greatest integer less than or equal to (n/2)).

There are $\left({n\atop n-2}\right)$ partitions of [n] into n-1 blocks.

Did I get this one correct?
I don't think you have the first one quite right. Consider what happens when n is even. How many partitions of [n] are there into sets of size n/2?

6. Originally Posted by awkward
I don't think you have the first one quite right. Consider what happens when n is even. How many partitions of [n] are there into sets of size n/2?
Yes, I saw the post about Stirling numbers of the second kind. The formula worked for 5 but not for 6. Funny that the problem was 3 sections ahead of Stirling numbers of the second kind!

### how many partitions of [n] into two blocks are there? how many partitions of [n] into n-1 blocks are there

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