As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

Saturday, May 1, 2010

Pascal Triangle - Ordered k-partitions

The number of integer solutions to
x1 + x2 + x3 = 6
with x1,x2,x3 >= 1 is equal to the number of integer solutions to
x1 + x2 + x3 = 3
with x1,x2,x3 >=0.
It is like distributing three similar balls into three different buckets, solutions are:
b b b | | ( x1=3, x2=0, x3=0 )
b b | b | ( x1=2, x2=1, x3=0 ).
A pattern appears. We are in fact looking at the number of different arrangements of three b's and 2 |'s, which is ( 3 + 2 )! / 3! 2! = C(5,2) = 10.
Let's list the solutions anyway.
3 - 0 - 0
2 - 1 - 0
2 - 0 - 1
1 - 2 - 0
1 - 1 - 1
1 - 0 - 2
0 - 3 - 0
0 - 2 - 1
0 - 1 - 2
0 - 0 - 3
or translated to the original question:

4 - 1 - 1
3 - 2 - 1
3 - 1 - 2
2 - 3 - 1
2 - 2 - 2
2 - 1 - 3
1 - 4 - 1
1 - 3 - 2
1 - 2 - 3
1 - 1 - 4.
They are in fact all the ordered 3-partitions of 6.
So we can interpret C(n,k) as the ordered (k+1)-partitions of (n+1).

Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)