False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)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: every weight substitution collapses the pattern inventory to the plain orbit count

Statement

False claim: no matter what weight function one chooses, the pattern inventory always collapses to the plain number of colour-orbits.

Facts & Assumptions

Given: the trivial action on the one-point set $X = {∙}$ with colour set $C = {blue, red}$ .

[L1]

Weighted pattern inventory evaluates the cycle index at the power sums of the colour weights (The weighted pattern inventory is the cycle index evaluated at the power sums).

Refutation

technique · direct

1.1

Give blue weight $1$ and red weight $u$ . Because the action is trivial and $X$ has one point, there are exactly two colouring orbits, with weights $1$ and $u$ . Hence the pattern inventory is $1 + u$ .

given

2.1

By [L1], the same conclusion is the cycle-index substitution for this one-point action. Unless $u = 1$ , the polynomial $1 + u$ is not the plain orbit count $2$ .

step 1.1L1

3.1

Therefore the displayed claim is false: a nonconstant weight assignment retains extra colour-profile information instead of collapsing to a single total.

step 2.1∎

Depends on

The weighted pattern inventory is the cycle index evaluated at the power sums

Used by

The substitution xᵢ=xⁱ can erase colour-profile information Counterexample

Dependency tree · two levels

5 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: Pólya's Inventory Theorem (standard reference, not scraped)