Partitions and Counting

Suppose that you have a set A with n elements. Is there a closed form solution of the number of ways that A can be partitioned in terms of n?

As an example, consider A = {a,b,c}. Then there are 5 different ways to partition this set.
1. a~a b~b c~c
2. a~b c~c
3. a~a b~c
4. a~c b~b
5. a~b~c

So I am trying to come up with some general formula for this. Even easier, there are 2 ways to partition a set with 2 elements, and only 1 way to partition a 1 element set.

#Combinatorics

Easy Math Editor

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Comments

I think you are referring to the Bell numbers:

https://en.wikipedia.org/wiki/Bell_number

http://mathworld.wolfram.com/BellNumber.html

And if I understand correctly, there are only 2 ways to partition a set with two elements.

Jon Haussmann - 4 years, 3 months ago

Thank you. I rushed in typing the last sentence, but I will edit that. Thank you! You found exactly what I was looking for.

Oli Hohman - 4 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$