Solution
= Solution
Define
$$
\tau(\sigma)(j)=n+1-\sigma(j).
$$
This is an involution on permutations. Every inequality is reversed, so it maps up-down permutations bijectively to down-up permutations.
= Solution
Define
$$
\tau(\sigma)(j)=n+1-\sigma(j).
$$
This is an involution on permutations. Every inequality is reversed, so it maps up-down permutations bijectively to down-up permutations.