Partitioning the Integers

Date: 15 Mar 1995 19:46:08 -0500
From: Anonymous
Subject: partitions

I'm a high school math teacher, and I recently assigned my discrete math 
students a project.  One of my students chose the topic of partitions of 
the positive integers.  We have studied various counting models, one of 
which is combinations with repetitions.  Are you aware of a book which 
treats the subject of partitions, or do you know of an elementary approach 
to the solution which I could explore with my students?  They know how to 
find, for example,  the number of four digit numbers, the sum of whose 
digits is 9, and I am convinced that the problem of partitioning the 
integers is related.

Thanks a lot!

Date: 16 Mar 1995 16:00:36 -0500
From: Stephen B Maurer) (by way of steve@mathforum.org Stephen Weimar
Subject: Re: partitions

First, it depends on exactly what you mean by partitions.  By the
partitions of 4, mathematicians usually mean

4
3  1
2  2
2  1  1
1  1  1  1

If order is considered, then the partitions are called ordered partitions.
Here they are for 4:

4
3  1
1  3
2  2
2  1  1
1  2  1
1  1  2
1  1  1  1

The latter objects are indeed closely related to combinations with
repetitions, but (unordered) partitions are a different story.  There are
no general, exact counting formulas for them, though there are many
asymptotic (limit) formulas, and many many theorems of the form "the number
of partitions of this form = the number of partitions of that form".  Any
book on number theory will have at least one chapter on partitions.
Combinatorics books typically spend less time on them.

Hope this helps.  Keep up the good work.

Stephen B Maurer
Professor of Mathematics
Swarthmore College