Usually we study the statistics of a permutation written in one row. Is there any result for the statistics of a permutation written in multiple rows? Let me give an example in order to be more clear: Let's put $1, 2, \cdots, 8$ into a $4\times2$ table and count the number of descents. Descents are defined to be a relationship that one number is greater than a number in the next column. If a number is greater than both of the numbers in the next column, we count twice. For example, $$\begin{pmatrix} 2 & 1 & 5 & 7 \\ 4 & 3 & 6 & 8 \end{pmatrix}$$ has descent number $3$, because $2>1, 4>1, 4>3$. For putting $2N$ numbers into two rows, I computed that the difference between the numbers of permutations with even descents and odd descents are: $2, 8, 144, 3200, 152320, 8672256, \cdots$, but I cannot figure out what this series is, and especially, what its exponential generating function is.
If there is only one row, I know the result is given by the Eulerian numbers and Eulerian polynimials. Are there generalized results for multiple rows?