Exam for the Night's Watch

Since the times of the Andals and the First Men, the function $f(p)$ has been known in Westeros, where $p$ is a permutation of arbitrary length. It is defined as follows.

Define the array $d$ as follows: d_i = |p_i - p_{i+1}| for all 1 \le i \le m-1, where $m$ is the length of the permutation. Then $f(p)$ is equal to the number of indices 1 \le i \le m-2 such that d_i \not\equiv d_{i+1} \pmod{2}.

Jon Snow has dreamed since childhood of joining the Night's Watch, so he is going to take the exam. Its essence is to find a permutation of length $n$ with the maximum possible value of $f(p)$. However, the examiner, Ser Alliser Thorne, hates Jon Snow and therefore requires that his permutation be lexicographically smallest among all permutations that achieve the maximum value of $f(p)$.

Since Jon Snow knows nothing, he needs your help. Please help him pass the exam for the Night's Watch.