Restoration of permutation

Today at school, Christopher learned about sequences and permutations. A permutation of numbers from $1$ to $n$ is a sequence a_1, \ldots, a_n, where each number from the set appears exactly once.

He found the following definitions particularly interesting: - A descent at position $i$ in the permutation \langle a_1, a_2, \ldots, a_n \rangle occurs when a_i > a_{i+1}. - A fixed point at position $i$ in the permutation \langle a_1, a_2, \ldots, a_n \rangle is when a_i = i.

Inspired by these concepts, he created a special type of permutation called the Robin permutation.

A permutation A = \langle a_1, a_2, \ldots, a_{2n} \rangle of $2n$ natural numbers from $1$ to $2n$ is a Robin permutation if it satisfies the following conditions: - $A$ has exactly $n$ descents, all occurring at odd positions (that is, a_{2i-1} > a_{2i} for all $i$); - $A$ has exactly $n$ fixed points.

For instance, the permutation \langle 3, 2, 6, 4, 5, 1 \rangle is a Robin permutation.

Christopher shared his discovery with Rabbit, who then proposed a transformation: remove all fixed points from the sequence and transform the remaining numbers into a permutation by replacing each number with the count of elements that do not exceed it in the permutation. For example, transforming the permutation \langle 3, 2, 6, 4, 5, 1 \rangle results in \langle 3, 6, 1 \rangle \rightarrow \langle 2, 3, 1 \rangle.

Now, Christopher wants to reconstruct a Robin permutation from the transformed permutation.