CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Positive-integer compositions have generating function $x / (1 - 2 x)$

Statement

Let $C$ be the class of compositions of positive integers, with size equal to the sum of the parts. Then

$OGF (C) = \frac{x}{1 - 2 x} .$

Facts & Assumptions

Given: The atomic class $Z$ and the constructions $SEQ (Z)$ and ${SEQ}_{\geq 1} (Z)$ .

[L1]

If $A$ has no size-zero objects then $SEQ (A)$ has generating function $1 / (1 - A (x))$ , and ${SEQ}_{\geq 1} (A)$ has generating function $A (x) / (1 - A (x))$ (If $A$ has no size-zero objects then $SEQ (A)$ has generating function $1 / (1 - A (x))$ ).

[L2]

Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions (Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions).

Proof

technique · direct

1.1

Let $P : = {SEQ}_{\geq 1} (Z)$ . Since $OGF (Z) = x$ , [L1] gives $OGF (P) = x / (1 - x)$ . An object of $P$ is a nonempty sequence of atoms, so it records one positive integer, namely its length.

2.1

A composition is a nonempty sequence of such positive-size blocks, so $C = {SEQ}_{\geq 1} (P)$ . Applying [L1] again gives $OGF (C) = OGF (P) / (1 - OGF (P)) = (x / (1 - x)) / (1 - x / (1 - x)) = x / (1 - 2 x)$ .

step 1.1L1L2algebra∎

Depends on

Used by

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

Robert Sedgewick and Kevin Wayne, Analysis of Algorithms, Section 3.9 (standard reference, not scraped)
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)

Positive-integer compositions have generating function x/(1−2x)x/(1−2x)