Compute the Eulerian number
Jelly, 8 bytes

Œ!Z&gt;2\Sċ


Try it online! (takes a while) or verify the smaller test cases.

How it works

Œ!Z&gt;2\Sċ  Main link. Arguments: n, m

Œ!        Generate the matrix of all permutations of [1, ..., n].
  Z       Zip/transpose, placing the permutations in the columns.
   &gt;2\    Compare columns pairwise with vectorizing greater-than.
          This generates a 1 in the column for each rise in that permutation.
      S   Compute the vectorizing sum of the columns, counting the number of rises.
       ċ  Count how many times m appears in the computed counts.