Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-27
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FALSE: Bell numbers count ordered set partitions

Statement

False claim: the Bell number Bn counts ordered set partitions of [n].

Facts & Assumptions

Given: The Bell numbers and ordered Bell numbers already discussed on this page.

Proof

technique · direct
1.1given

For n=3, The Stirling numbers of the second kind and the Bell numbers gives B3=5.

2.1step 1.1givenalgebra

But the ordered set partitions of [3] are counted by 1⋅1!+3⋅2!+1⋅3!=13. Since 5≠13, the Bell numbers do not count ordered set partitions.

3.1step 2.1∎

Therefore the claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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