I have $N$ lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has $N-1$ possibilities and the second one has $N-3$ as each one cannot make connections itself and the first one has already formed one. So the total possibility is $$(N-1)(N-3)(N-5)(N-7).....$$ and so on. Now I want to impose two conditions.
Case(i): If I impose a condition that each site cannot be connected with the nearest neighbor, how many ways I can make the connections. How do complete this counting problem with this condition?
Case(ii): Apart from the above condition(immediate neighbors should not be connected), If I impose a further condition that each site can be connected with other or it can also be left unconnected. How do I count the number of ways doing this?
I know both case(i) and case(ii) will have different answers. I really don't know where to start this problem at all