CorollaryStatement: Literature-sourcedProof: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Distinct-part product generating function

Statement

In $Z ⟦ x ⟧$ ,

$\sum_{n \geq 0} p_{d i s t} (n) x^{n} = \prod_{m \geq 1} (1 + x^{m}) .$

Facts & Assumptions

Given: one abstract object $u_{m}$ of size $m$ for each integer $m \geq 1$ .

[F1]

The empty partition has distinct parts, and a nonempty partition into distinct parts chooses each positive part size at most once (The functions p(n), p_k(n), and the standard restricted partition families).

[L1]

If a combinatorial class has one object of each positive size $m$ in a permitted layer, then its powerset construction contributes the factor $(1 + x^{m})$ at size $m$ (If $A$ has no size-zero objects then $PSET (A)$ has generating function $\prod_{n \geq 1} (1 + x^{n})^{a_{n}}$ ).

Proof

technique · direct

1.1F1construct

By [F1], a partition into distinct parts is exactly a subset of the set ${u_{1}, u_{2}, \dots}$ : include $u_{m}$ when the part $m$ occurs, and omit it otherwise. The total size of the chosen subset is the sum of the selected part sizes.

2.1step 1.1L1∎

Applying [L1] to the class with one object $u_{m}$ in each positive size $m$ gives one factor $(1 + x^{m})$ for every $m \geq 1$ . Step 1.1 identifies the resulting powerset objects with distinct-part partitions, so their generating function is $\prod_{m \geq 1} (1 + x^{m})$ .

Depends on

Used by

Dependency tree · two levels

8 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

Stephen Melczer, An Invitation to Enumeration, Chapter 9: Integer Partitions (standard reference, not scraped)