Alphabeta MathHow 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 permutations of tabulated by inversions, descents, major index and excedances
Example
For , the four statistics take the following values.
| permutation | inv | des | maj | exc |
|---|---|---|---|---|
| 0 1 2 3 | 0 | 0 | 0 | 0 |
| 0 1 3 2 | 1 | 1 | 3 | 1 |
| 0 2 1 3 | 1 | 1 | 2 | 1 |
| 0 2 3 1 | 2 | 1 | 3 | 2 |
| 0 3 1 2 | 2 | 1 | 2 | 1 |
| 0 3 2 1 | 3 | 2 | 5 | 1 |
| 1 0 2 3 | 1 | 1 | 1 | 1 |
| 1 0 3 2 | 2 | 2 | 4 | 2 |
| 1 2 0 3 | 2 | 1 | 2 | 2 |
| 1 2 3 0 | 3 | 1 | 3 | 3 |
| 1 3 0 2 | 3 | 1 | 2 | 2 |
| 1 3 2 0 | 4 | 2 | 5 | 2 |
| 2 0 1 3 | 2 | 1 | 1 | 1 |
| 2 0 3 1 | 3 | 2 | 4 | 2 |
| 2 1 0 3 | 3 | 2 | 3 | 1 |
| 2 1 3 0 | 4 | 2 | 4 | 2 |
| 2 3 0 1 | 4 | 1 | 2 | 2 |
| 2 3 1 0 | 5 | 2 | 5 | 2 |
| 3 0 1 2 | 3 | 1 | 1 | 1 |
| 3 0 2 1 | 4 | 2 | 4 | 1 |
| 3 1 0 2 | 4 | 2 | 3 | 1 |
| 3 1 2 0 | 5 | 2 | 4 | 1 |
| 3 2 0 1 | 5 | 2 | 3 | 2 |
| 3 2 1 0 | 6 | 3 | 6 | 2 |
Facts & Assumptions
Given: The inversion generating function, the major-index generating function, and the equidistribution of descents and excedances on (The inversion generating function of is , The major-index generating function of is , Descents and excedances are equidistributed on ).
Verification
The table is an exhaustive enumeration of the one-line permutations of , with each statistic computed directly from its definition.
Reading off the distributions from the table gives for inversions, the same polynomial for the major index, and for both descents and excedances. This matches the three cited theorems.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
10 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.