DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Descent sets, major index, excedances and fixed points of a permutation

Definition

Let $n \in N$ and let $σ = [σ (0), \dots, σ (n - 1)] \in S_{n}$ .

The descent set of $σ$ is

$Des (σ) : = {i \in {0, \dots, n - 2} : σ (i) > σ (i + 1)},$

and the descent number is

$des (σ) : = ∣ Des (σ) ∣ .$

Its major index is

$maj (σ) : = \sum_{i \in Des (σ)} (i + 1) .$

The shift by $1$ is the convention fixed in This page uses the published inversion number and the shifted Eulerian indexing $A (n, k)$ .

The excedance set of $σ$ is

$Exc (σ) : = {i \in n : σ (i) > i},$

and the excedance number is

$exc (σ) : = ∣ Exc (σ) ∣ .$

For later use, the anti-excedance set is

$AExc (σ) : = {i \in n : σ (i) < i},$

and the fixed-point set is

$Fix (σ) : = {i \in n : σ (i) = i} .$

Depends on

This page uses the published inversion number and the shifted Eulerian indexing $A(n,k)$

Used by

Two permutations can have the same descent set and different inversion numbers Counterexample
Eulerian numbers and Eulerian polynomials Definition
FALSE: the major index equals the inversion number for every permutation False statement
Under Foata's fundamental transformation, anti-excedances become descents Lemma
Descents and excedances are equidistributed on Sₙ Theorem
Foata's transformation sends major index to inversion number Theorem

Dependency tree · two levels

3 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)
Andrew Lin, MIT 18.212 Algebraic Combinatorics, Lecture 10 (standard reference, not scraped)