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 ${\displaystyle \textstyle \left\langle {n \atop k}\right\rangle }$.

The Eulerian polynomials ${\displaystyle A_{n}(t)}$ are defined by the exponential generating function

The Eulerian numbers ${\displaystyle A(n,k)}$ may also be defined as the coefficients of the Eulerian polynomials:

An explicit formula for ${\textstyle A(n,k)}$ is

A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common characteristics with Pascal's triangle. Values of ${\textstyle A(n,k)}$ (sequence A008292 in the OEIS) for ${\textstyle 0\leq n\leq 9}$ are:

For larger values of ${\textstyle n}$, ${\textstyle A(n,k)}$ can also be calculated using the recursive formula