I'm trying to develop a program that evaluates all possible arrangements (combinations of series and parallel) of k resistors from a set.
From what I've read, if the resistors were identical values, then this would just be a permutation formula. But what if the k resistors have different values? Say I don't care if any arrangements end up having the same net resistance as each other, I just want to know the total number of ways I can arrange R1, R2, ... Rk.
Ignoring the set for the moment (or setting N = k), I get that when k =1, there's only one arrangement (obviously). When k=2, there are 2 (both in series, both in parallel).
When k = 3, I get 8:
1 x all three in series
1 x all three in parallel
3 x 1 in parallel with 2 in series, and
3 x 1 in series with 2 in parallel
When k becomes 4 or above, this becomes a lot more complicated. 3 series resistors can be parallel with 1, or 1 can be in series with 3 parallel, or 2 parallel resistors can be in series with 2 other parallel resistors, or 2 series can be in series with 2 parallel, etc.
Does anybody have a solution for this problem, or know of any statistical formula that would help me?
EDIT: after a quick and dirty attempt to sketch out all possible arrangements for 4 resistors, I counted 52 different arrangements. As best as I can explain:
1 x all 4 in series
1 x all 4 in parallel
4 x 1 in series with 3 in parallel
12 x 1 in series with 1 resistor in parallel with 2 others in series
6 x 2 in series, in series with 2 in parallel
3 x 2 in parallel in series with 2 in parallel
3 x 2 in series parallel with 2 in series
6 x 2 in series in parallel with 2 in parallel
4 x 1 in parallel with 3 in series
12 x 1 in parallel with (1 in series with 2 in parallel)
That's probably not very easy to follow, but I'm pretty sure I'm correct. So now I have for N=k,
k = 1: y = 1
k = 2: y = 2
k = 3: y = 8
k = 4: y = 52
k = 5: y = ??