TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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

Statement

For every $n \in N$ ,

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

Facts & Assumptions

Given: A natural number $n$ .

[L1]

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

[L2]

The inversion number is $inv (σ) = ∣ Inv (σ) ∣$ (Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations).

Proof

technique · constructive

1.1

For $σ \in S_{n}$ with Lehmer code $L (σ) = (ℓ_{1}, \dots, ℓ_{n})$ , the inversion number is $inv (σ) = ℓ_{1} + \dots + ℓ_{n}$ : the component $ℓ_{i}$ counts exactly the inversions whose left entry is the position $n - i$ .

L2algebra

2.1

By [L1] and step 1.1, $\sum_{σ \in S_{n}} q^{inv (σ)} = \sum_{(ℓ_{1}, \dots, ℓ_{n})} q^{ℓ_{1} + \dots + ℓ_{n}} = \prod_{i = 1}^{n} \sum_{j = 0}^{i - 1} q^{j} = \prod_{i = 1}^{n} [i]_{q} = [n]_{q}!$ .

step 1.1L1algebra

3.1

This is the claimed generating function identity. At $n = 0$ , both sides are $1$ .

step 2.1∎

Depends on

Used by

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.

Sources

Richard P. Stanley, Enumerative Combinatorics, Volume 1, second edition (standard reference, not scraped)
Felix Gotti, MIT 18.211 Combinatorial Analysis, Lecture 5 (standard reference, not scraped)

The inversion generating function of SnSn​ is [n]q![n]q​!