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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 Eulerâ€™s number $$e$$ (the base of the natural logarithm), or with the Euler numbers $$E_n$$ (defined by $$1/\cosh(x) = \sum_{n=0}^\infty (E_n/n!) x^n$$).