TITLE: A question about certain sets of permutations of the ordered pairs $(1,1),(1,2),\cdots,(1,n),\cdots,(n,1),(n,2),\cdots,(n,n)$
QUESTION [15 upvotes]: Let $n>1$ be a given positive integer. For any $0\leq k\leq n^2$, let $A_k$ be the set of permutations $((i_1,j_1),(i_2,j_2),\cdots,(i_{n^2},j_{n^2}))$ of the ordered pairs $(1,1),(1,2),\cdots,(1,n),\cdots,(n,1),(n,2),\cdots,(n,n)$ satisfying $i_1\leq i_2\leq \cdots\leq i_k$ and  $j_{k+1}\leq j_{k+2}\leq \cdots\leq j_{n^2}$.
For any $$((i_1,j_1),(i_2,j_2),\cdots,(i_k,j_k),(i_{k+1},j_{k+1}),\cdots,(i_{n^2},j_{n^2}))\in A_k,$$ we have $$((j_{k+1},i_{k+1}),(j_{k+2},i_{k+2}),\cdots,(j_{n^2},i_{n^2}),(j_1,i_1),(j_2,i_2),\cdots,(j_k,i_k))\in A_{n^2-k}.$$
So it is easy to see that $\vert A_k\vert=\vert A_{n^2-k}\vert$ for any $0\leq k\leq [n^2/2]$. 

Question: Do we have that $\vert A_0\vert<\vert A_1\vert<\cdots<\vert A_{[n^2/2]}\vert$?

REPLY [6 votes]: This is not an answer, but perhaps a relevant thought. The combinatorial description of $A_k$, for $0\le k\le n^2$, together with the Gamma integral $\Gamma(n+1)=\int_0^{\infty}x^ne^{-x}dx$ leads to the generating function
$$\sum_{k=0}^{n^2} A_kt^k=\int_0^\infty\cdots \int_0^{\infty}e^{-\sum_{i=1}^n(x_i+y_i)}\prod_{i,j=1}^n(x_i+ty_j)dx_1dy_1\cdots dx_ndy_n.$$
A polynomial whose coefficients form a unimodal (resp. log-concave) sequence is called a unimodal (resp. log-concave) polynomial. It is known that a polynomial with real roots is log-concave and thus unimodal. So unimodality holds for the polynomial being integrated above. Unfortunately log-concavity or unimodality are not preserved under linear combinations so one can't conclude the desired property just from the integral representation above. Maybe someone with knowledge in analysis will find this wording more useful.