TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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The exponential formula gives the Bell-number generating function

Statement

Let $u$ be a second formal indeterminate. In $Q [u] ⟦ x ⟧$ , the block-count-refined exponential generating function of set partitions is

$\sum_{n \geq 0} \sum_{k = 0}^{n} S (n, k) u^{k} \frac{x^{n}}{n!} = \exp ⁣ (u (e^{x} - 1)) .$

In particular,

$\sum_{n \geq 0} B_{n} \frac{x^{n}}{n!} = \exp (e^{x} - 1) .$

Facts & Assumptions

Given: The second-kind definition of The Stirling numbers of the second kind and the Bell numbers and the labelled symbolic rules of The labelled constructions translate into the usual exponential-generating-function rules.

Proof

technique · direct

1.1

A set partition is a labelled set of nonempty labelled sets. Marking each block by a factor of $u$ replaces the basic block EGF $e^{x} - 1$ by $u (e^{x} - 1)$ . Applying the labelled-set rule therefore gives $\sum_{n \geq 0} \sum_{k = 0}^{n} S (n, k) u^{k} \frac{x^{n}}{n!} = \exp ⁣ (u (e^{x} - 1)) .$

given

2.1

Setting $u = 1$ sums over all block counts and therefore replaces $\sum_{k = 0}^{n} S (n, k)$ by $B_{n}$ . This yields $\sum_{n \geq 0} B_{n} x^{n} / n! = \exp (e^{x} - 1)$ .

step 1.1given

3.1

Steps 1.1 and 2.1 prove the refined formula and its Bell-number specialization.

step 1.1step 2.1∎

Depends on

Used by

Standard labelled specializations give involutions, ordered Bell numbers, and partitions without singletons Corollary

Dependency tree · two levels

9 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

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics — Symbolic Combinatorics (standard reference, not scraped)
Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.4 (standard reference, not scraped)