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 $q$ are called edge-invariant if $e_p = e_q$. How many edge-invariant walks of length $l$ exist, 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 $2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158$ for $l = 1,\dots,13$. It seems to be related to k-abelian equivalence classes, where the general problem seems to be rather difficult, but maybe the 2-abelian cases is simpler?