DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Eulerian numbers and Eulerian polynomials

Definition

For $n \in N$ and $k \in N$ , the Eulerian number $A (n, k)$ is the number of permutations of $S_{n}$ with exactly $k$ descents:

$A (n, k) : = ∣ {σ \in S_{n} : des (σ) = k} ∣ .$

Thus $A (n, k) = 0$ unless $0 \leq k \leq n - 1$ when $n \geq 1$ , and we set $A (0, 0) = 1$ .

The Eulerian polynomial is

$A_{n} (t) : = \sum_{k = 0}^{n - 1} A (n, k) t^{k} (n \geq 1),$

with $A_{0} (t) : = 1$ .

Depends on

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

Used by

The Eulerian numbers satisfy A(n,k)=A(n,n-1-k) Corollary
The Eulerian-polynomial exponential generating function in ℚ(t)⟦ x⟧ Corollary
The Eulerian numbers satisfy A(n,k)=(k+1)A(n-1,k)+(n-k)A(n-1,k-1) Theorem
Worpitzky's identity for n≥1 Theorem

Dependency tree · two levels

2 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 13 (standard reference, not scraped)