ExampleConstruction: AI-adaptedVerification: Not suppliedaudited 2026-07-31

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

Example

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

$μ (A, B) = (- 1)^{∣ B ∖ A ∣} (A \subseteq B)$

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

Equivalently, the comparable pairs split as follows:

| $∣ B ∖ A ∣$ | number of pairs | $μ (A, B)$ | |---:|---:|---:| | $0$ | $8$ | $1$ | | $1$ | $12$ | $- 1$ | | $2$ | $6$ | $1$ | | $3$ | $1$ | $- 1$ |

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: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ vanish when $x < y$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)

The full Möbius table of the Boolean lattice 2[3]2[3]