="Robert Munafo" <mrob27@gmail.com>
The first amount to be subtracted is 2^(2^(N-1)-1).
Strictly syntactically, from the exponentiation, and given that the formula for A003645 in the OEIS is sum((-1)^k*binomial(n, k)2^2^(n-k), k=0..n)/2 it seems like you might be close to the right enumeration neighborhood.
I think the cube labeling is the most natural way to visualize this one, but I'd love to hear of something better.
Me too, but this is more than I was able to come up with! You might want to verify that any proposed "model schema" also down-scales to the 5 cases for n=2.
I looked at the OEIS entry but don't know what a "covering" is and didn't bother looking at the references or anything else (-:
I hear ya. But "covering"'s not too hard to visualize. Suppose you have a pallet of N adjacent wine barrels that you want to keep rain water out of. And what you have at hand are 2^N-1 distinct cheap thin plastic N-barrel covers, each covering with a different combination of closed barrelheads and open tops . How many ways, using those covers, can you ensure that no barrel remains uncovered? Another related way to view 109 is that it's the number of ways to distinctly partition 7 using IOR for addition. Alas none of that seems to suggest a natural visualization.