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For questions about the Eulerian numbers $A_{n,k}$, defined as the number of permutations in the symmetric group $S_n$ having $k$ descents. Not to be confused with Euler’s number $e$ or the Euler numbers $E_n$.

Let $S_n$ denote the symmetric group on $n$ letters, more specifically, the set of all bijections from $[n] := \{ 1,\dotsc,n \}$ to itself. For a permutation $\pi \in S_n$, let the descent set of $\pi$ be defined as $$\operatorname{Des}(\pi) := \{ i \in [n-1] : \pi(i) > \pi(i+1) \}.$$ Let $\operatorname{des}(\pi) := \lvert \operatorname{Des}(\pi)\rvert$. The Eulerian number $A_{n,k}$ is defined as $$A_{n,k} := \{ \pi \in S_n : \operatorname{des}(\pi) = k \},$$ that is, $A_{n,k}$ counts the number of permutations in $S_n$ having exactly $k$ descents.

One can define the related notion of ascent of a permutation in a similar manner. The number of permutations with $k$ descents equals the number of permutations with $k$ ascents (consider the complement map $\pi \mapsto \pi^c$ where $\pi^c(i) = n + 1 - \pi(i)$), so one can also define $A_{n,k}$ to be the number of permutations in $S_n$ with $k$ ascents.

The Eulerian polynomial $A_n(x)$ is defined by $$A_n(x) = \sum_{k=0}^n A_{n,k} x^k.$$ These satisfy nice recursions, and are important objects of study in combinatorics.

Do not confuse the Eulerian numbers $A_{n,k}$ with Euler’s number $e$ (the base of the natural logarithm), or with the Euler numbers $E_n$ (defined by $1/\cosh(x) = \sum_{n=0}^\infty (E_n/n!) x^n$).

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