Given a complete graph $(V,E)$ with $n$ vertices $V$ and walks $p \in V^{l+1}$ of length $l$. We say the edges of walks $p$ are the multiset 

$$
e_p = \{ (p_i,p_{i+1}) \mid 1 \leq i \leq l \}.
$$  
Now, two walks $p$ and $q$, where $q$ is a permutation of $p$, are called edge-invariant if $e_p = e_q$. How many different walks of length $l$, which are not edge-invariant, exist? More precisely, i.e., 
$$
max_{P \subset V^{l+1}} | \{ P \mid \forall p, q \in P: e_p \neq e_q \textrm{ for } p \neq q \}|.
$$

For example, $n = 2$, it's $4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158$ for $l = 1,\dots,12$. Specifically for $l = 3$, out of the 16 possible walks, two are edge-invariant:
$$
e_{[1,1,2,1]} = e_{[1,2,1,1]} \textrm{ and } e_{[2,2,1,2]} = e_{[2,1,2,2]}.
$$

For $n = 3$, [it's given here][1] as $9, 27, 75, 186, 414, 840, 1578, 2784, 4662, 7476, 11556$ for $l = 1,\dots,11$.


It seems to be related to [k-abelian equivalence classes][2], where the general problem seems to be rather difficult, but maybe the 2-abelian cases is simpler? 


  [1]: https://oeis.org/A289658
  [2]: https://arxiv.org/abs/1605.03319