RemarkRemark: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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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.
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The cycle-index necklace count agrees with the published CYC count

Remarks

Necklace count from the cyclic-group cycle index derives the necklace formula by averaging fixed colourings under the rotation action of $C_{n}$ .

The earlier published item The number of necklaces of length $n$ on an $m$ -letter alphabet is $\frac{1}{n} \sum_{d ∣ n} φ (d) m^{n / d}$ reaches the same sequence through the symbolic $CYC$ construction. The two derivations therefore agree term by term:

$# {necklaces of length n on m colours} = \frac{1}{n} \sum_{d ∣ n} φ (d) m^{n / d} .$

The agreement matters because the cycle-construction route and the cycle-index route spend different machinery, but they count the same orbit set.

Depends on

Used by

Two-colour necklaces of length 6 by the published CYC formula Example

Dependency tree · two levels

6 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

Ben Lynn, Polya Theory: The Cycle Index Polynomial (standard reference, not scraped)