CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A family with infinitely many objects of size $2$ is not a combinatorial class

Counterexample

Let $A : = {a_{0}, a_{1}, a_{2}, \dots}$ and define $∣ a_{j} ∣ : = 2$ for every $j \in N$ . Then the level $A_{2}$ is infinite, so $A$ is not a combinatorial class.

Facts & Assumptions

Given: A combinatorial class is required to have finite size- $n$ levels for every $n$ (Combinatorial classes, counting sequences and ordinary generating functions).

Verification

technique · direct

1.1

The size- $2$ level of the displayed family is $A_{2} = {a_{0}, a_{1}, a_{2}, \dots}$ , which is infinite.

given

2.1

This violates the defining finiteness condition on levels, so the family is not a combinatorial class.

step 1.1given∎

Depends on

Combinatorial classes, counting sequences and ordinary generating functions

Used by

Nothing in the library uses this result yet.

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.

A family with infinitely many objects of size 22 is not a combinatorial class