The eulerian2(n, k) command counts the number of permutations ${\mathrm{pi}}_1{\mathrm{pi}}_2\mathrm{...}{\mathrm{pi}}_{2n}$ of the multi-set $\left[1\,1\,2\,2\,\mathrm{...}\,n\,n\right]$ having two properties:

All numbers between the two occurrences of $m$ are greater than $m$.

There are k ascents, namely, k places where ${\mathrm{pi}}_j<{\mathrm{pi}}_{j+1}$.

This function can be computed via the recurrence

Second-order Eulerian numbers are important because of their connection with Stirling numbers. For integers $m$ and $0\le n$ we have:

$\left|\mathrm{Stirling1}\left(m\&comma;m-n\right)\right|$ = ${\left(\mathrm{-1}\right)}^n\mathrm{Stirling1}\left(m\&comma;m-n\right)$ = $\sum _{k=0}^n\mathrm{eulerian2}\left(n\&comma;k\right)\left(\genfrac{}{}{0ex}{}{m+k}{2n}\right)$

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

$\mathrm{Matrix}\left(\left[\mathrm{seq}\left(\left[\mathrm{seq}\left(\mathrm{eulerian2}\left(n\&comma;k\right)\&comma;k=0..5\right)\right]\&comma;n=0..5\right)\right]\right)$

$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 1& 2& 0& 0& 0& 0\\ 1& 8& 6& 0& 0& 0\\ 1& 22& 58& 24& 0& 0\\ 1& 52& 328& 444& 120& 0\end{array}\right]$

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