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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer and let $\\pi$ be a [[permutation|0BQMWCNS8S]] of $\\set{1, \\ldots, n}$.\nA *descent* of $\\pi$ is an index $i$ with $1 \\le i \\le n - 1$ and $\\pi(i) \u003e \\pi(i + 1)$.\nFor an integer $k$, the *Eulerian number* $A(n, k)$ is the number of permutations of $\\set{1, \\ldots, n}$ with exactly $k$ descents.\n\n## Remarks\n\nExample: for $n = 3$, the permutations in one-line notation are $123$, $132$, $213$, $231$, $312$ and $321$.\nTheir numbers of descents are $0, 1, 1, 1, 1, 2$, so $A(3, 0) = 1$, $A(3, 1) = 4$ and $A(3, 2) = 1$.\n\nNon-example: the index $i = 3$ is not a descent of $\\pi = 2\\,4\\,1\\,3$, because $\\pi(3) = 1 \u003c 3 = \\pi(4)$.\nThe descents of $\\pi$ are only $i = 2$, so $\\pi$ is counted by $A(4, 1)$.\n\nOther notations are $\\left\\langle {n \\atop k} \\right\\rangle$ and $E(n, k)$.\nThe numbers satisfy $A(n, k) = A(n, n - 1 - k)$ and $\\sum_{k} A(n, k) = n!$.\nThey obey the recurrence $A(n, k) = (k + 1) A(n - 1, k) + (n - k) A(n - 1, k - 1)$.\nAn explicit formula is\n$$A(n, k) = \\sum_{j=0}^{k+1} (-1)^j \\binom{n + 1}{j} (k + 1 - j)^n.$$\nWorpitzky's identity states that $x^n = \\sum_{k=0}^{n-1} A(n, k) \\binom{x + k}{n}$ for $n \\ge 1$.\n",
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