---
tag: 5PS2496830
kind: concept
name: Eulerian number
msc: [05A05, 11B83, 05A15]
sources:
  - url: https://en.wikipedia.org/wiki/Eulerian_number
same_as:
  wikidata: Q1373849
  wikipedia: Eulerian number
---

## Summary

The Eulerian number $A(n, k)$ counts the [[permutations|05RWFKVTJ7]] of $\set{1, \ldots, n}$ with exactly $k$ descents, see the [[definition|B2EMR1MA1H]].
For each $n \ge 1$ the numbers $A(n, 0), \ldots, A(n, n-1)$ add up to $n!$.
They are the coefficients of the Eulerian polynomials, which appear in the sums $\sum_{m \ge 0} m^n t^m$.
They should not be confused with the Euler numbers or with Eulerian paths in graphs.

## Explanations

### L1 human

Write a permutation as a row of numbers and mark each place where the next number is smaller.
These places are the descents.
For $n = 3$, the row $1\,2\,3$ has no descent, the row $3\,2\,1$ has two, and the other four rows have exactly one.
So the Eulerian numbers for $n = 3$ are $1, 4, 1$.