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The eulerian1(n, k) command counts the number of permutations of ${\mathrm{pi}}_1{\mathrm{pi}}_2\mathrm{...}{\mathrm{pi}}_n$ of $\left\{1\&comma;2\&comma;\mathrm{...}\&comma;n\right\}$ that have k ascents, namely, k places where ${\mathrm{pi}}_j<{\mathrm{pi}}_{j+1}$.

This function can be computed via the recurrence

It also satisfies the following identities:

$\mathrm{eulerian1}\left(n\&comma;k\right)$ = $\sum _{m=0}^k{\left(\mathrm{-1}\right)}^m\left(\genfrac{}{}{0ex}{}{n+1}{m}\right){\left(k+1-m\right)}^n$ = $\sum _{m=0}^n{\left(\mathrm{-1}\right)}^{n-m-k}m!\left(\genfrac{}{}{0ex}{}{n-m}{k}\right)\mathrm{Stirling2}\left(n\&comma;m\right)$

$\mathrm{with}\left(\mathrm{combinat}\right)\&colon;$

$\mathrm{Matrix}\left(\left[\mathrm{seq}\left(\left[\mathrm{seq}\left(\mathrm{eulerian1}\left(n\&comma;k\right)\&comma;k=0..5\right)\right]\&comma;n=0..5\right)\right]\right)$

$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0\\ 1& 4& 1& 0& 0& 0\\ 1& 11& 11& 1& 0& 0\\ 1& 26& 66& 26& 1& 0\end{array}\right]$

R.L. Graham, D.E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, Reading, Mass., 1989.