partition function
- From: niklaus@xxxxxxxxx
- Date: Mon, 6 Oct 2008 10:21:11 -0700 (PDT)
i'm trying to find the number of non negative integral solutions of
the equation a + 2b + 3c + 4d = n ,i find it is equation to the
partition of n into at most 4 parts. Wikipedia says this is the
definition of partition .
Can some one help me understand why is the number of non integral
solutions to the above equation is equal to number of partions atmost
4 of n.
i could write the above equation as
(a+b+c+d) + (b+c+d) + (c+d) + d = n , if we replace a+b+c+d as x1, b+c
+d as x2 , c+d as x3 then we have x1 +x2 +x3 + d = n which is number
of partitions of n into 4 parts exactly but with below constraints.
0<=d<=n/4 , 0<=c<=n/4 , 0<=b<=n/2 , 0<=a<=n, if there weren't this
constraint i could directly substitute x1,x2,x3, x4 and then say it is
4 parts exactly.
now again we can write a+2b +3c=n if d=0, then we have number of
solutions of n into 3 parts, if c ,d=0, n into 2 parts, b=c=d=0,
then n into 1 part,
I'm stuck here . how do i proceed . This looks easy but i'm unable to
find the insight.
.
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