9. Pieces of Numbers

We will spend some time learning about a very simple new concept - one that only requires addition. We’ll explore it.

9.1. Pieces of Numbers – The Simplest Addition

Here is a question nearly every human could answer.

How many ways can you divide a pile of 5 sticks into (non-empty) piles?

Let’s see.

There is the trivial one where you don’t divide it at all.  5=5. There is the one where you just make five piles of one stick each.  1+1+1+1+1=5. You could have four in one pile.  That leaves 4+1=5. But if you have three in one pile, there are two ways to do the other two.

3+2=5. 3+1+1=5.

Almost done!  I guess the only other possibility is if there are piles of 2 and 1.

2+2+1=5. 2+1+1+1=5.

That makes seven ways in total.  We call these things integer partitions.  As George Andrews says in this delightful little book: “An integer partition is a way of splitting a number into integer parts.”  More precisely, a partition of a nonnegative integer n is a representation of n as a sum of positive integers, called summands or parts of the partition. The order of the summands is irrelevant.

The function giving the number of partitions of n is called p(n).  So our example shows that p(5)=7. Amazingly, there are real physics applications of this! See this list of articles and this more dubious example.  A typical quote: “The number partitioning problem can be interpreted physically in terms of a thermally isolated noninteracting Bose gas trapped in a one-dimensional harmonic-oscillator potential.”

9.1.1. Exploration 1

Try to compute p(n) for n\leq 10. Divide up this work in groups! Maybe three or four people should work on each one, and then cross-check their work.

9.1.2. Exploration 2

The question to explore is:

  • Partitions of :math:`n` using only odd parts. This is notated :math:`p(n mid text{ odd parts })`.

9.1.3. Exploration 3

The question to explore is:

  • Partitions of :math:`n` using only even parts, or :math:`p(n mid text{ even parts })`.

9.1.4. Exploration 4

The question to explore is:

  • Partitions of :math:`n` using distinct parts; that is, all the numbers in the partition are different.

Try these out for small (!) n. See if you find any possible patterns. Again, it is probably best if some larger groups work together and split up the work to make it more efficient.