The full Möbius table of the Boolean lattice $2^{[3]}$

Let $[3]=\{1,2,3\}$ and order $2^{[3]}$ by inclusion (The Boolean lattice of subsets of a finite set and its rank levels). The complete table is determined by

$\mu(A,B)=(-1)^{|B\setminus A|}\qquad(A\subseteq B)$

(For $A\subseteq B$ in a finite Boolean lattice, $\mu(A,B)=(-1)^{\lvert B\setminus A\rvert}$). Thus the value is $1$ on the diagonal, $-1$ when $B$ adds one element, $1$ when it adds two elements, and $-1$ from $\varnothing$ to $[3]$.

Equivalently, the comparable pairs split as follows:

| $|B\setminus A|$ | number of pairs | $\mu(A,B)$ | |---:|---:|---:| | $0$ | $8$ | $1$ | | $1$ | $12$ | $-1$ | | $2$ | $6$ | $1$ | | $3$ | $1$ | $-1$ |

For a cover $A\subset A\cup\{i\}$ the recurrence reads $1+(-1)=0$. For the top interval it reads $1+3(-1)+3(1)+(-1)=0$, in agreement with The Möbius recurrence: $\mu_P(x,x)=1$ and both interval sums of $\mu_P$ vanish when $x<y$.