Question

Distinct sums and entropy

Answer 1

I don't think you are computing the p\_UV-like value properly.

For U = {1,2} and V = {3,4}, row by row you have:

(-1,-1,16,17,17,18) (1,1,18,17,17,16) (-1,-1,16,17,17,18) (1,1,18,17,17,16)

So H = 1.

And for U' = {1} and V' = {3,4}:

 (-1,16,17) (1,18,17) (-1,16,17) (1,18,17)

And H = 1 too.

Note, they are speaking of sequences, not sets. And indeed, if U' and V' are resp. subsets of U and V then you are only getting subsequences so the entropy will be smaller.