CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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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.

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The Bell numbers satisfy $B_{n + 1} = \sum_{k = 0}^{n} (\binom{n}{k}) B_{k}$

Statement

For every $n \in N$ ,

$B_{n + 1} = \sum_{k = 0}^{n} (\binom{n}{k}) B_{k} .$

Facts & Assumptions

Given: A partition of $[n + 1]$ , counted by The Stirling numbers of the second kind and the Bell numbers.

Proof

technique · direct

1.1given

Let $S$ be the block containing $n + 1$ , and let $T : = [n] ∖ (S ∖ {n + 1})$ . Then $T$ is exactly the set of elements not lying in the distinguished block. If $∣ T ∣ = k$ , there are $(\binom{n}{k})$ choices for $T$ , and after that the elements of $T$ may be partitioned arbitrarily in $B_{k}$ ways.

1.2givenconstruct

Conversely, every choice of a subset $T \subseteq [n]$ and a partition of $T$ determines a unique partition of $[n + 1]$ : put all elements of $[n] ∖ T$ together with $n + 1$ into one block and keep the chosen partition of $T$ for the other blocks.

2.1step 1.1step 1.2∎

Summing over all possible values $k = ∣ T ∣$ gives the claimed recurrence for $B_{n + 1}$ .

Depends on

The Stirling numbers of the second kind and the Bell numbers

Used by

Touchard's congruence: for prime p, Bₙ₊ₚ≡ Bₙ+Bₙ₊₁ (mod p) Corollary

Dependency tree · two levels

3 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

Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.6 (standard reference, not scraped)
Combinatorics Through Guided Discovery, Bell numbers (standard reference, not scraped)

The Bell numbers satisfy Bn+1=∑k=0n(nk)BkBn+1​=∑k=0n​(kn​)Bk​