Counting non-degenerate parallelograms in permutation arrays I am interested in finding out more about the function $P(n)$ that counts the number of distinct non-degenerate parallelograms whose vertices are in distinct rows and columns of an $n$ x $n$ array.  Enumerative C++ code with nested loops shows that $P(4)=6$, $P(5)=62$, and $P(6)=362$ and also produced the values of $P(n)$ for $4 \leq n \leq 129$.  This data (or a heuristic argument) shows that $P(n)$ has $n^6$ growth.  If possible, I would like to find a recurrence relation or generating function for $P(n)$. Curve fitting for large $n$ appears to suggest that there is an asymptotic constant $\alpha \approx 0.0514$ such that $P(n)\sim \alpha n^6$.  I would also like to know if $\alpha$ exists and to know its true value if it does exist.
Theorems 1 and 2 in the new paper  

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p44

that I coauthored solve this problem.
Theorems 1 and 2 in the new paper  

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p44

that I coauthored solve this problem.