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?

Update: The original problem seems too broad and too hard. Let me specific to the case which is enough for my purpose. It seems permutations with even descents are always more than those with odd descents. If it is hard to determine exactly what is the difference between them, is there a way to pair each odd descent permutation with a unique even descent permutation?

  • $\begingroup$ It seems to be quite easy to guess in the case with n rows, with two entries in each row. You get some sort of q-binomial as generating function, times n!^2. $\endgroup$ – Per Alexandersson Jun 7 '18 at 19:30

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