Sitting younger, older alternately.

Once the youngest has sat, the other seven people form a line of alternating age comparisons: $$1<A>B<C>D<E>F<G>1$$ It is well-known that the number of ways to arrange the other seven people in such a line is given by the up/down number $A_7=272$. The Wikipedia page linked describes a simple recurrence for these numbers.