Eulerian number

Remarks

Example: for $n = 3$, the permutations in one-line notation are $123$, $132$, $213$, $231$, $312$ and $321$. Their numbers of descents are $0, 1, 1, 1, 1, 2$, so $A(3, 0) = 1$, $A(3, 1) = 4$ and $A(3, 2) = 1$.

Non-example: the index $i = 3$ is not a descent of $\pi = 2\,4\,1\,3$, because $\pi(3) = 1 < 3 = \pi(4)$. The descents of $\pi$ are only $i = 2$, so $\pi$ is counted by $A(4, 1)$.

Other notations are $\left\langle {n \atop k} \right\rangle$ and $E(n, k)$. The numbers satisfy $A(n, k) = A(n, n - 1 - k)$ and $\sum_{k} A(n, k) = n!$. They obey the recurrence $A(n, k) = (k + 1) A(n - 1, k) + (n - k) A(n - 1, k - 1)$. An explicit formula is $$A(n, k) = \sum_{j=0}^{k+1} (-1)^j \binom{n + 1}{j} (k + 1 - j)^n.$$ Worpitzky's identity states that $x^n = \sum_{k=0}^{n-1} A(n, k) \binom{x + k}{n}$ for $n \ge 1$.