# total number of different partitions of a set having four elements is

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How many different partitions with exactly two parts can be made of the set {1,2,3,4}? There are 4 elements in this list that need to be partitioned into 2 parts. I wrote these out and got a total ...

## How many different partitions with exactly n parts can be made of a set with k-elements?

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How many different partitions with exactly two parts can be made of the set {1,2,3,4}? There are 4 elements in this list that need to be partitioned into 2 parts. I wrote these out and got a total of 7 different possibilities:

{{1},{2,3,4}} {{2},{1,3,4}} {{3},{1,2,4}} {{4},{1,2,3}} {{1,2},{3,4}} {{1,3},{2,4}} {{1,4},{2,3}}

Now I must answer the same question for the set {1,2,3,...,100}. There are 100 elements in this list that need to be partitioned into 2 parts. I know the largest size a part of the partition can be is 50 (that's 100/2) and the smallest is 1 (so one part has 1 number and the other part has 99). How can I determine how many different possibilities there are for partitions of two parts without writing out extraneous lists of every possible combination? Can the answer be simplified into a factorial (such as 12!)?

Is there a general formula one can use to find how many different partitions with exactly n parts can be made of a set with k-elements?

mathdiscrete-mathematicsdata-partitioning

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edited Sep 3, 2012 at 17:18

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asked Feb 16, 2012 at 17:54

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## 2 Answers

5

1) stackoverflow is about programming. Your question belongs to https://math.stackexchange.com/ realm.

2) There are 2n subsets of a set of n elements (because each of n elements may either be or be not contained in the specific subset). This gives us 2n-1 different partitions of a n-element set into the two subsets. One of these partitions is the trivial one (with the one part being an empty subset and other part being the entire original set), and from your example it seems you don't want to count the trivial partition. So the answer is 2n-1-1 (which gives 23-1=7 for n=4).

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answered Feb 16, 2012 at 18:05

penartur 9,6845 5 gold badges 38 38 silver badges 48 48 bronze badges

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Feb 16, 2012 at 22:31

@BlueRaja-DannyPflughoeft Yes, i'm aware of it :) What i wasn't aware of is that one can use html tags like^{ when writing answers. –}

penartur

Feb 17, 2012 at 9:41

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The general answer for n parts and k elements would be the Stirling number of the second kind S(k,n).

Please beware that the usual convention is with n the total number of elements, thus S(n,k)

Computing the general formula is quite ugly, but doable for k=2 (with the common notation) :

Thus S(n,2) = 1/2 ( (+1) * 1 * 0n +(-1) * 2 * 1n + (+1) * 1 * 2n ) = (0-2+2n)/2 = **2n-1-1**

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## How to find all partitions of a 4

Answer: As one of the comments suggested, you can use the Stirling numbers of the second kind - Wikipedia, S(n,k), to calculate the number of ways to separate n objects into k partitions. So S(4,1)=1 is the number of ways to put 4 objects into 1 partition, S(4,2)=7 is the number of ways to have 2...

How do I find all partitions of a 4-element set?

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Sort Glenn Redd

B.S. in Mathematics, Kennesaw State University (Graduated 2016)Author has 1.9K answers and 2.3M answer views4y

As one of the comments suggested, you can use the Stirling numbers of the second kind - Wikipedia

, S(n,k), to calculate the number of ways to separate n objects into k partitions. So S(4,1)=1 is the number of ways to put 4 objects into 1 partition, S(4,2)=7 is the number of ways to have 2 partitions, S(4,3)=6 is 3 partitions, and S(4,4)=1 is 4 partitions. So the sum of these 1+7+6+1=15 is the number of total possible partitions of a 4 element set.

This is also the 5th Bell number (since the 1st bell number is the number of partitions for 0 elements, that means 4 elements is the 5th), but one of the ways to calculate the Bell number is to add up all of the Stirling numbers of the second kind for the given number of elements, so…

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Sam Sinai

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What is a partition of a set?

> What is a partition of a set?

A Partition of a set [ https://en.m.wikipedia.org/wiki/Partition_of_a_set ],

A A

, is a set of subsets,

B i Bi , of A A such that: ⋃ i B i =A ⋃iBi=A and ∀i≠j: B i ∩ B j =∅ ∀i≠j:Bi∩Bj=∅

That is it is a splitting of the set into disjoint subsets that cover the original set. Every member of the original set is in one and only one subset.

You should be familiar with a partition from your experiences at school. Your year (the set of people your age at your school) was probably partitioned into classes. You were in one...

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How many subsets does a set with 4 elements have?

2^4 = 16. The empty set, {A}, {B}, {C}, {D}, {A, B}, {A, C}, {A, D}, }B, C}, {B, D}, {C, D}, {A, B, C}, {A, B, D}, {A,C, D}, {B, C, D}, and {A, B, C, D} itself.

Generally, to construct a subset, list all elements of the set and to each element assign either YES (belongs to the subset) or NO (does not belong to the subset). This can be done in 2 ways for each element; therefore, if the original set has n elements, the total number of possible choices is 2*2*2*…*2 (n times), i.e. 2^n.

James Fullwood

PhD in Mathematics, Florida State University (Graduated 2012)Author has 367 answers and 971K answer views4y

Related

How is it possible to have a set with one element only?

Consider for example the set of all planets you've been to. My guess is this set has a single element.

Rohini Ramachandran Upvoted by

Samuel Gomes da Silva

, Ph.D. Mathematics & Set Theory, University of São Paulo (2004) and

Dwight House

, Ph. D. Mathematics, Duke University (1972)3y

Related

Why is the empty set considered a set?

My little brother had trouble with the concept of empty sets when he was first learning about them in high school. This is how I explained it to him (this might not be entirely mathematical, kindly correct me if anything I say is not logical):

A set is like a box with some stuff in it. It is a well defined collection of objects. When you look inside the box, you should be able to tell if something’s in it or not, there should be no ambiguity.

Now let’s consider an empty set. It’s a set with nothing in it. So, it’s like an empty box. A box is still a box even if there’s nothing in it!

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Related

Can an element be a set?

Depending on your set theory, it’s possible that it can’t be anything else.

Most mathematicians use ZFC as their set theory. In ZFC, essentially everything is a set. There are some things that aren’t sets (informally called “classes”), but those things can’t be elements.

There are set theories in which there are things that are not themselves sets, but can still be elements of sets. These are generally called “ur-elements”.

Sets can always be elements. The complicated question is whether an element can be something other than a set.

Dan Grubb

Ph. D. in Mathematics, Kansas State University (Graduated 1986)Upvoted by

David Joyce

, Ph.D. Mathematics, University of Pennsylvania (1979)Author has 185 answers and 59.2K answer viewsMar 6

Related

If a set A has an infinite number of elements, what is its cardinality?

This is like asking what the cardinality is for a non-empty set.

## If a set A has 4 elements then total number of proper subsets of set A is

If a set A has 4 elements then total number of proper subsets of set A is

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Guys, does anyone know the answer?