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.

Odd-part product generating function

Statement

In $Z ⟦ x ⟧$ ,

$\sum_{n \geq 0} p_{o d d} (n) x^{n} = \prod_{m \geq 1} (1 - x^{2 m - 1})^{- 1} .$

Facts & Assumptions

Given: one abstract object $v_{m}$ of size $2 m - 1$ for each integer $m \geq 1$ .

[F1]

The empty partition has odd parts, and a nonempty partition into odd parts may use each odd part size with arbitrary multiplicity and uses no even part size (The functions p(n), p_k(n), and the standard restricted partition families).

[L1]

If a combinatorial class has one object in each permitted positive size, its multiset construction contributes the geometric factor $(1 - x^{d})^{- 1}$ for each allowed size $d$ (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

1.1F1construct

By [F1], a partition into odd parts is exactly a multiset of the objects $v_{1}, v_{2}, \dots$ : the multiplicity of $v_{m}$ records how often the odd part $2 m - 1$ occurs. The total size of the multiset is the sum of the odd parts.

2.1step 1.1L1∎

Applying [L1] to the class ${v_{1}, v_{2}, \dots}$ gives the product $\prod_{m \geq 1} (1 - x^{2 m - 1})^{- 1}$ . Step 1.1 identifies its multiset objects with partitions into odd parts, so this is the generating function for $p_{o d d} (n)$ .

Depends on

Used by

Euler's theorem by generating functions Theorem

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)