The dinner table problem: the rectangular case

$n$ people are seated randomly at a rectangular table with $\lfloor n/2\rfloor$and$\lceil n/2\rceil$ seats along the two opposite sides for two dinners. What’s the probability that neighbors at the first dinner are no more neighbors at the second one? We give an explicit formula and we show that its asymptotic behavior as $n$ goes to infinity is $e^{-2}(1+4\mathrm{/}n)$ (it is known that it is $e^{-2}(1-4\mathrm{/}n)$ for a round table). A more general permutation problem is also considered.