I formulated this question while thinking about resistor networks.
Suppose you are given N distinct resistors. How many ways are there to combine them into a resistor network?
A resistor network is one of the following:
- a single resistor
- two resistor networks connected in parallel
- two resister networks connected in series
The two later cases are commutative and associative. ie A+B = B+A and (X+Y)+Z = X+(Y+Z). Equal networks should only be counted once.
By what method can we calculate the number of combinations?