## Partitions of Graphs I: Partitions of Whole Numbers

I’ve been meaning to blog about this for some time.  I guess that I had just better do it.  But it will be in pieces.  A little at a time.  That’s the way blogs are supposed to be right?

Let’s say you give me a whole number, like 5.  A partition of 5 is some other whole numbers that add up to 5.  For instance 4+1 is a partition of 5.  OK, everyone knows that 4+1=5.  So what’s the game we are playing here?  We are counting 4+1 and something else like 3+2 as different things.  Never mind that they equal the same thing, we are going to count them as different.  However, we want to count 4+1 as the same as 1+4.  We also want 3+2 to be the same as 2+3.

Ok, those are the basic rules.  Matheticians like to count things, so let’s count how many partions of 5 there are:

5 (did I mention that this one counts even though there isn’t a plus sign in it?)

4+1

3+2

3+1+1

2+2+1

2+1+1+1

1+1+1+1+1

That’s all of them (how do I know that? . . .).  You can check yourself.  There are seven.  Let’s do another one.  The partitions of 4 are:

4

3+1

2+2

2+1+1

1+1+1+1

There are five of them.

Try some yourself.

Ok, this seems pretty stupid.  2+2=4, duh. But counting how many partitions each whole number has is a big deal. There’s lots of beautiful math going on here and this is an active area of current research.  The partitions of whole numbers touches lots of other areas of math in strange ways.

So what do I have to say about partitions of whole numbers?  Not much today, except that I find them fascinating and a while ago I decided I wanted to relate them to another family of intriguing mathematical objects, what mathematicians call graphs and non-mathematicians usually call networks . . .What those are will be the subject of my next post.  Stay tuned.

Part II: Graphs