The ménage problem (statement)

Setup. Arrange $n$ men around a round table ($(n-1)!$ ways), then seat $n$ women in the gaps (alternating). No couple adjacent.

IE. Total arrangements (men fixed): $n!$ seatings of women. Bad: couple $i$ adjacent. IE gives
$$M_n=\sum_{k=0}^n(-1)^k\frac{2n}{2n-k}\binom{2n-k}{k}(n-k)!.$$

Conclusion. The ménage numbers are given by the IE formula above. $M_3=1$, $M_4=2$, $M_5=13$.