Consider the TSP problem. Let $F_n=\{\text{all cyclic permutations $\pi$ on $n$ objects} \}$. Define the neighborhood 2-change of TSP as
$$N_2^n(f)=\{g: g\in F_n \text{ and $g$ can be obtained from $f$ as follows:}\\ \quad\quad\quad\quad\quad\quad\quad\;\;\;\; \text{remove two edges from the tour; then}\\ \quad\quad\quad\quad\quad\;\, \text{replace them with two edges}\}.$$
What is the cardinality of $N_2^n(f)$?
I start with few examples like $N_2^1(f)=0$, $N_2^2(f)=0$, $N_2^3(f)=3$ but I cannot find it in general.