ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

The Lehmer codes of $S_{4}$ recover $[4]_{q}!$

Example

For $S_{4}$ , the Lehmer-code codomain is

${0} \times {0, 1} \times {0, 1, 2} \times {0, 1, 2, 3} .$

Grouping these $24$ code vectors by the sum of their coordinates gives

$1 + 3 q + 5 q^{2} + 6 q^{3} + 5 q^{4} + 3 q^{5} + q^{6},$

which is $[4]_{q}!$ .

Facts & Assumptions

Given: The Lehmer code is a bijection $S_{4} \to {0} \times {0, 1} \times {0, 1, 2} \times {0, 1, 2, 3}$ (The Lehmer code is a bijection $S_{n} \to \prod_{i = 1}^{n} {0, \dots, i - 1}$ ).

[L1]

The inversion generating function of $S_{4}$ is $[4]_{q}!$ (The inversion generating function of $S_{n}$ is $[n]_{q}!$ ).

Verification

technique · direct

1.1givenalgebra

Every code vector has the form $(0, a, b, c)$ with $a \in {0, 1}$ , $b \in {0, 1, 2}$ , and $c \in {0, 1, 2, 3}$ , so there are $24$ of them. Counting by the sum $a + b + c$ gives the coefficient sequence $1, 3, 5, 6, 5, 3, 1$ .

2.1step 1.1L1∎

Therefore $\sum_{σ \in S_{4}} q^{inv (σ)} = 1 + 3 q + 5 q^{2} + 6 q^{3} + 5 q^{4} + 3 q^{5} + q^{6}$ , which matches [L1].

Depends on

Used by

Nothing in the library uses this result yet.

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.

The Lehmer codes of S4S4​ recover [4]q![4]q​!