Eulerian number

Summary

The Eulerian number $A(n, k)$ counts the permutations of $\set{1, \ldots, n}$ with exactly $k$ descents, see the definition. 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.