The Dinner Table Problem: The Rectangular Case

Consider n people who are seated randomly at a rectangular table with ⌊n/2⌋ and ⌈n/2⌉ seats along the two opposite sides, for two dinners. What is the probability that neighbors at the first dinner are no longer neighbors at the second one? We give an explicit formula and show that its asymptotic behavior as n goes to infinity is e −2 (1 + 4/n) (it is known that it is e −2 (1−4/n) for a round table). A more general permutation problem is also considered.