# Thread: Partitions proof

1. ## Partitions proof

Let the number of partitions of n into parts each less than or equal to m be denoted P(n,m). Prove
P(n,m) = P(n, m-1) + P(n-m,m)

2. ## Re: Partitions proof

Originally Posted by BlinkyBoo
Let the number of partitions of n into parts each less than or equal to m be denoted P(n,m). Prove
P(n,m) = P(n, m-1) + P(n-m,m)
I have written on this question. So I tell you whoever wrote your question has left out some cases.
What happens if $\displaystyle m=1~?$ What happens if $\displaystyle m=n~?$

Here is the MathCad way of doing it.
[ATTACH=CONFIG]23869[/ATTACH]

3. ## Re: Partitions proof

what is mathcad? >.> ... i'm not really following that equation
I'm a little confused because there is a formula in the book about partitioning n into exactly r parts where order counts = C(n-1,r-1)
but then there's another one that says "the number of ways of writing n as the sum of r or fewer integers where the order of the summands matters is C(r+n-1,n)"
but i didn't think they applied because order shouldn't count, should it?
I was also trying to play with the concept that the number of partitions of n into r or less parts = the number of partitions of n into parts that are r or less, but have gotten nowhere so far :\

Originally Posted by Plato
$m=1~?$ What happens if $m=n~?$
if m = 1 there should only be one partition, n itself, right?
and if m = n, then all the partitions = 1

4. ## Re: Partitions proof

could you be more specific please?

5. ## Re: Partitions proof

Originally Posted by BlinkyBoo
could you be more specific please?
You can study this web page.