Consider a partial order on piles of 2 black stones and 2 white stones. Say that one configuration of pile of stones,A, is smaller than another,B, if you can join piles of A together to get those of B. The smallest configuration under this order is ◦|◦|•|• while the largest is ◦◦•• .
so i drew the hasse diagram corresponding with this problem, now i must compute the mobius function of the form $\mu(◦|◦|•|•,A)$ for this partial order for all configurations of A. How would i compute one mobius function, since im assuming theres seven to be computed. I have more to add, i just have it in a rough draft.
Now i realized i am missing an element from my hasse diagram, and how it will effect the current structure i have ( ◦•|◦•).
this new image is what i came up with to include the element i was missing and now i am more stuck than i was before because i apparently repeated the same thing three times in the third row ( ◦•|•|◦)
It seems, you are only asked to calculate the value of $\mu$ on the given pair of elements (an interval), where $A$, I guess, wants to denote the bottom element, the one-pile for $A$.
Also, I'm not 100% sure that the order to consider is not just the opposite of what you wrote.
About the correspondence your picture suggests: nobody claimed that there is such a one-to-one correspondence with the subsets of $\{1,2,3\}$. Nevertheless, it's very nice that you found it, and it is indeed almost full. Actually, this is an injective lattice homomorphism.
And, by the way, the $\{0\}$ at the bottom should be $\emptyset$.