Eulerian number

In combinatorics, the Eulerian number ${\textstyle A(n,k)}$ is the number of permutations of the numbers 1 to ${\textstyle n}$ in which exactly ${\textstyle k}$ elements are greater than the previous element (permutations with ${\textstyle k}$ "ascents").

Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis.

The polynomials presently known as Eulerian polynomials in Euler's work from 1755, Institutiones calculi differentialis, part 2, p. 485/6. The coefficients of these polynomials are known as Eulerian numbers.

Other notations for ${\textstyle A(n,k)}$ are ${\textstyle E(n,k)}$ and $\textstyle \left\langle {n \atop k}\right\rangle$ .