Commuting Functions

Two functions $\mathbf{f}$ and $\mathbf{g}$ (\mathbf{f}, \mathbf{g}: \mathbf{X} \rightarrow \mathbf{X}) are commuting if and only if for every $\mathbf{x}$, \mathbf{f}(\mathbf{g}(x)) = \mathbf{g}(\mathbf{f}(x)). For example, the functions \mathbf{f}(x) = x+1 and \mathbf{g}(x) = x-2 are commuting, while the functions \mathbf{f}(x) = x+1 and \mathbf{g}(x) = 2x are not commuting.

Each function $\mathbf{h}$ (\mathbf{h}: N_n \rightarrow N_n, where N_n = \{1, 2, \ldots, n\} and $n$ is a positive integer) can be represented as a value list—a list in which the $i$-th element is equal to $\mathbf{h}(i)$. For example, the function \mathbf{h}(x) = \lfloor x/2 \rfloor from N_5 to N_5 has the value list [1, 1, 2, 2, 3].

Value lists are ordered lexicographically: the list [a_1, \ldots, a_n] is smaller than the list [b_1, \ldots, b_n] if and only if there exists an index $k$ such that a_k < b_k, and a_l = b_l for all indices l < k.

The function $\mathbf{f}$ (\mathbf{f}: X \rightarrow X) is called bijective if for every $y$ in $X$, there is exactly one $x$ in $X$ such that \mathbf{f}(x) = y.

Given a bijective function $\mathbf{f}$ (\mathbf{f}: N_n \rightarrow N_n, where $n$ is a positive integer), find a function $\mathbf{g}$ that commutes with $\mathbf{f}$ and has the lexicographically smallest possible value list.