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

Statement

Let $I_{n}$ be the number of involutions of $[n]$ , let $F_{n}$ be the number of ordered set partitions of $[n]$ , and let $N_{n}$ be the number of set partitions of $[n]$ having no singleton block. The labelled symbolic method gives the following exponential generating functions:

involutions: $\sum_{n \geq 0} I_{n} \frac{x^{n}}{n!} = \exp ⁣ (x + \frac{x^{2}}{2});$
ordered Bell numbers: $\sum_{n \geq 0} F_{n} \frac{x^{n}}{n!} = \frac{1}{2 - e^{x}};$
set partitions with no singleton blocks: $\sum_{n \geq 0} N_{n} \frac{x^{n}}{n!} = \exp (e^{x} - 1 - x) .$

Facts & Assumptions

Given: The labelled symbolic rules and the set-partition formulas already proved on this page.

Proof

technique · direct

1.1given

An involution is a labelled set of fixed points and transpositions, so its EGF is $\exp (x + x^{2} / 2)$ .

1.2given

An ordered Bell structure is a sequence of nonempty labelled sets. The EGF of one nonempty labelled set is $e^{x} - 1$ , so the sequence rule gives $1 / (1 - (e^{x} - 1)) = 1 / (2 - e^{x})$ .

1.3given

A partition with no singleton blocks is a labelled set of blocks of size at least $2$ . Subtracting the singleton contribution $x$ from the basic block EGF $e^{x} - 1$ leaves $e^{x} - 1 - x$ , and the set rule gives $\exp (e^{x} - 1 - x)$ .

2.1step 1.1step 1.2step 1.3∎

These are exactly the three claimed specializations.

Depends on

Used by

Ordered Bell numbers already differ from Bell numbers at n=3 Example
The involutions of [4] and their cycle decomposition Example
The set partitions of [4] with no singleton blocks Example
FALSE: Bell numbers count ordered set partitions False statement

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Sources

Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.8 (standard reference, not scraped)
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics — Symbolic Combinatorics (standard reference, not scraped)