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. He first studied them in 1749 (though first printed in 1768).

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