ExampleConstruction: AI-adaptedVerification: Not suppliedSession-authored (Fable 5 assisted)audited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Möbius table of a four-element chain

Example

For the chain $c_0<c_1<c_2<c_3$ , the complete upper-triangular table is

$\mu(c_i,c_j)$	$j=0$	$j=1$	$j=2$	$j=3$
$i=0$	$1$	$-1$	$0$	$0$
$i=1$		$1$	$-1$	$0$
$i=2$			$1$	$-1$
$i=3$				$1$

This is On a finite chain, the Möbius function is $1$ on the diagonal, $-1$ on covers and $0$ on longer intervals: diagonal entries are $1$ , cover entries are $-1$ , and entries spanning more than one cover are $0$ . For example, the recurrence gives $\mu(c_0,c_2)=-(1-1)=0$ , then $\mu(c_0,c_3)=-(1-1+0)=0$ (The Möbius recurrence: $\mu_P(x,x)=1$ and both interval sums of $\mu_P$ vanish when $x<y$ ).

Depends on

Used by

A four-element chain and a four-element diamond have equal-size endpoint intervals but Möbius values 0 and 1 Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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