Thread: Help with integer partitions

1. Help with integer partitions

I'm still new to integer partition and I've gotten this far.
The number of order independant ways you can partition an integer into a set of numbers whose sum is N is defined as P(n).
Some examples are P(5)=7, P(8)=22 , and P(10)=42.

Let P(n,k) be the number of partitions in which each member is at least k. Also this is congruent to the number of partitions with at least k members.

$P(n,k) = 0\ if \ k>n$
$P(n,k) = 1\ if\ k=n$
$P(n,k) = P(n,k+1) + P(n-k,k)$

If I say P(10,5) that is the set of all partitions of 10 whose members are all >=5.
Which in this case (10,5)=2. 10 and 5+5.

Is there a way to find the partitions of X in which all the members <=K ? I've been trying to figure out some way to add and/or subtract various P(n,k) values to achieve this but i'm at a lost.

I did see some pages on the generating functions of P(n), but I haven't fully understood them yet. Am I going to be forced to use the generating function to get what I want? Namely the number of Partitions of X whose members are all <= to some K

2. Re: Help with integer partitions

Do you know the 'little book' MATHEMATICS OF CHOICE by Ivan Niven?
There is a very nice discussion of this topic in that book.
I have programmed MathCad to produce a table of partitioning an integer N into k or fewer summands. The recursion used is similar to the one you posted.