How many equivalence classes does the following equivalence relation (over permutations) have?

Question

Let $n\in\mathbb{N}$. Consider the set of all permutations over $n$. Two permutations $\pi_1 = i_1..i_n$ and $\pi_2 = j_1..j_n$ are not equivalent if either $i_n i_1$ appears in $\pi_2$ (that is, $\exists k: j_k=i_n \land j_{k+1}=i_1$) or $j_n j_1$ appears in $\pi_1$. How many equivalence classes there are?

Answer 1

Your question cannot be answered because it assumed a false premise. The operation you described is not an equivalence relation, because it violates transitivity:

$$(1, 2, 3, 4) = (4, 3, 2, 1) = (3, 4, 1, 2)$$ $$(1, 2, 3, 4) \neq (3, 4, 1, 2)$$