False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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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.
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FALSE: conjugation itself is the distinct-parts to odd-parts bijection

Statement

False claim: conjugation sends every partition with distinct parts to a partition into odd parts.

Facts & Assumptions

Given: the distinct-part partition $λ = (4, 2, 1)$ .

[L1]

Conjugation is the transpose of the Ferrers diagram (Conjugating a partition twice returns the original partition).

[L2]

The true bijection between odd-part and distinct-part partitions is Glaisher's map (Glaisher's bijection between odd-part and distinct-part partitions).

Refutation

technique · direct

1.1

The Ferrers diagram of $(4, 2, 1)$ has column lengths $3, 2, 1, 1$ , so [L1] gives $(4, 2, 1)^{'} = (3, 2, 1, 1)$ .

givenL1

2.1

The conjugate partition $(3, 2, 1, 1)$ has an even part, namely $2$ , so it is not a partition into odd parts.

step 1.1

3.1

Therefore conjugation does not send every distinct-part partition to an odd-part partition, and the displayed claim is false. By [L2], the correct bijection is a different map.

step 2.1L2∎

Depends on

Used by

Conjugating (4,2,1) does not produce an odd-part partition Counterexample

Dependency tree · two levels

4 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)