TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)judge 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.

Glaisher's bijection between odd-part and distinct-part partitions

Statement

For every integer $n \geq 0$ , there is a bijection between partitions of $n$ into odd parts and partitions of $n$ into distinct parts.

Facts & Assumptions

Given: an integer $n \geq 0$ .

[F1]

Partitions into odd parts and partitions into distinct parts are the two families counted by $p_{o d d} (n)$ and $p_{d i s t} (n)$ (The functions p(n), p_k(n), and the standard restricted partition families).

Proof

technique · bijection

1.1

Let $λ$ be a partition of $n$ into odd parts. For each odd integer $m$ , let $c_{m}$ be its multiplicity in $λ$ , and write the binary expansion

$c_{m} = \sum_{j \geq 0} ε_{m, j} 2^{j}, ε_{m, j} \in {0, 1} .$

Define $G (λ)$ to contain the part $2^{j} m$ once for every pair $(m, j)$ with $ε_{m, j} = 1$ . Since

$\sum_{j \geq 0} ε_{m, j} 2^{j} m = m c_{m},$

the parts with odd core $m$ have the same total $m c_{m}$ before and after the replacement. Summing over the odd cores shows that the total sum of the parts of $G (λ)$ is still $n$ . [F1, construct, algebra]

1.2

Conversely, let $μ$ be a partition of $n$ into distinct parts. Write each part uniquely as $2^{j} u$ with $u$ odd, and replace it by $2^{j}$ copies of the odd part $u$ . The resulting partition $H (μ)$ has only odd parts and still sums to $n$ .

F1construct

2.1

The parts of $G (λ)$ are distinct. Indeed, every positive integer has a unique expression $2^{j} u$ with $u$ odd, so two equal parts in $G (λ)$ would come from the same odd core $u$ and the same power $2^{j}$ , hence from the same binary digit.

step 1.1

3.1

The two constructions are inverse. Starting from $λ$ , step 1.2 reconstructs from each part $2^{j} m$ of $G (λ)$ exactly the $2^{j}$ copies of $m$ encoded by the corresponding binary digit, so $H (G (λ)) = λ$ . Starting from $μ$ , step 1.1 groups together all parts with the same odd core and reassembles exactly the original powers of two, so $G (H (μ)) = μ$ . Therefore $G$ is a bijection.

step 1.1step 2.1step 1.2∎

Depends on

The functions p(n), p_k(n), and the standard restricted partition families

Used by

Dependency tree · two levels

3 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

Darij Grinberg, Enumerative Combinatorics: class notes (standard reference, not scraped)
Stephen Melczer, An Invitation to Enumeration, Chapter 9: Integer Partitions (standard reference, not scraped)