How to compute a function on a set of natural numbers using recursion - matlab

Question

I am working on a property of a given set of natural numbers and it seems difficult to compute. I build a function 'fun' which takes two inputs, one is the cardinal value and another is the set. If the set is empty then fun should return 0 because fun depends on the product of the set and fun on all subsets of the complement set.
For clarification here is an example:
S is a set given S={1,2,3,4}. The function fun(2,S) is defined as
fun(2,S)=prod({1,2})*[fun(1,{3}) + fun(1,{4}) + fun(2,{3,4})] +
prod({1,3})*[fun(1,{2}) + fun(1,{4}) + fun(2,{2,4})] +
prod({1,4})*[fun(1,{3}) + fun(1,{2}) + fun(2,{2,3})] +
prod({2,3})*[fun(1,{4}) + fun(1,{1}) + fun(2,{1,4})] +
prod({2,4})*[fun(1,{1}) + fun(1,{3}) + fun(2,{3,1})] +
prod({3,4})*[fun(1,{1}) + fun(1,{2}) + fun(2,{1,2})]
prod is defined as the product of all elements in a set, for example
prod({1,2})=2;
prod({3,2})=6
I am trying to compute the function fun using recursive method in MATLAB but it's not working. The base case is the cardinal value should be more than zero that means there should be at least one element in the set other wise prod will be zero and fun will return zero.
Update Pseudo code:
fun(i,S)
if |S|=1 && i!=0
return prod(S)
else if i==0
return 0
else
prod(subset s', s' is a subset of S and |s'|=i)*(sum over fun((for i=1 to m),{S-s'}), m=|S-s'|) //I don't know how to write code for this part and need help.
end if
end fun
prod(s)
n=|s|
temp=1
for i=1 to n
temp *=s(i) //s(1) is the 1st element of s
end for
return temp
end prod
Thanks.

Answer 1

With the pseudo code you added to your question it's nearly impossible to implement the function. Everything is put into one line which is incomplete (at least the outer sum is missing).
1) Formalize your algorithm in a way it can be used to implement. The following pseudo code is probably not correct because I don't exactly know what you want, but it should give an idea how to do it.
fun(i,S)
if i==0
return 0
else if |S|=1
return S
else
r=0
for s1 in subsets of S with size i
f=0
for s2 in subsets of setdiff(S,s') with size <=i
f=f+fun(s2,|s2|)
end
r=r+prod(s1)*f
end for
end if
end fun
2) use arrays [1,2,3,4] instead of cells {1,2,3,4}
3) prod is a built-in function, no need to reimplement it.