CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-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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The major-index generating function of $S_{n}$ is $[n]_{q}!$

Statement

For every $n \in N$ ,

$\sum_{σ \in S_{n}} q^{maj (σ)} = [n]_{q}! .$

Facts & Assumptions

Given: Foata's transformation and the inversion generating function of $S_{n}$ .

[L0]

Foata's transformation is a bijection of $S_{n}$ (Foata's transformation is a bijection of $S_{n}$ ).

[L1]

Foata's transformation sends major index to inversion number (Foata's transformation sends major index to inversion number).

[L2]

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

Proof

technique · direct

1.1

By [L0], Foata's transformation is a bijection of $S_{n}$ , so reindexing the sum over $S_{n}$ by $Φ$ and using [L1] gives $\sum_{σ \in S_{n}} q^{maj (σ)} = \sum_{σ \in S_{n}} q^{inv (σ)}$ .

L0L1

2.1

Apply [L2] to the right-hand side.

step 1.1L2∎

Depends on

Used by

The number of excedances is Eulerian but not Mahonian Counterexample
The permutations of S₄ tabulated by inversions, descents, major index and excedances Example
FALSE: the major index equals the inversion number for every permutation False statement

Dependency tree · two levels

9 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

Richard P. Stanley, Enumerative Combinatorics, Volume 1, second edition (standard reference, not scraped)

The major-index generating function of SnSn​ is [n]q![n]q​!