CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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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Integer partitions have generating function $\prod_{n \geq 1} (1 - x^{n})^{- 1}$

Statement

Let $P$ be the class of integer partitions, with size equal to the sum of the parts. Then

$OGF (P) = \prod_{n \geq 1} (1 - x^{n})^{- 1} .$

Facts & Assumptions

Given: For each $n \geq 1$ , a single abstract object $p_{n}$ of size $n$ , and the combinatorial class $U : = {p_{1}, p_{2}, \dots}$ .

[L1]

If $A$ has no size-zero objects then $OGF (MSET (A)) = \prod_{n \geq 1} (1 - x^{n})^{- a_{n}}$ (If $A$ has no size-zero objects then $MSET (A)$ has generating function $\prod_{n \geq 1} (1 - x^{n})^{- a_{n}}$ ).

Proof

technique · direct

1.1construct

A multiset of objects from $U$ records exactly an integer partition: the multiplicity of $p_{n}$ is the number of parts equal to $n$ , and the total size is the sum of the parts. Also $U$ has exactly one object of each positive size and none of size $0$ .

2.1step 1.1L1∎

Applying [L1] with $a_{n} = 1$ for every $n \geq 1$ gives $OGF (P) = \prod_{n \geq 1} (1 - x^{n})^{- 1}$ .

Depends on

Used by

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 (standard reference, not scraped)
Stephen Melczer, An Invitation to Enumeration, Chapter 5: Combinatorial Constructions (standard reference, not scraped)

Integer partitions have generating function ∏n≥1(1−xn)−1∏n≥1​(1−xn)−1