ExampleConstruction: AI-adaptedVerification: Not suppliedaudited 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

$μ (c_{i}, c_{j})$	$j = 0$	$j = 1$	$j = 2$	$j = 3$
$i = 0$	$1$	$- 1$	$0$	$0$
$i = 1$		$1$	$- 1$	$0$
$i = 2$			$1$	$- 1$
$i = 3$				$1$

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 $μ (c_{0}, c_{2}) = - (1 - 1) = 0$ , then $μ (c_{0}, c_{3}) = - (1 - 1 + 0) = 0$ (The Möbius recurrence: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ 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 · two levels

10 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)