Given a set of $n$ states $V = \{ s_1, s_2, \ldots, s_n \}$, and a complete digraph $G = (V, A)$ where $A = \{ (a,b) \mid (a,b) \in V^2\; \text{and}\; a \neq b \}$, I'm interested in finding cyclic trails through $G$ which traverse every element of $A$ exactly once.
I've found a number of different ways to construct such trails, including one based on A097285 (the sequence that contains exactly once every pair $(i,j)$ of distinct positive integers): $t = (a_1, a_2, \ldots, a_{n*(n+1)})$ where $a_i = (s_{A097285(i)}, s_{A097285(i+1)})$.
Given that there's more than one way to construct such trails, I'd like to better understand how to construct such trails, how many there are for a given $n$, and how the properties of different trails compare (e.g. in my practical application of this, it's desirable to construct trails which visit each vertex at least once as early as possible along the trail).
Also, what other problems are related to or isomorphic to this? I'm kinda getting vibes of Gray codes and modular arithmetic, but I've not been able to figure out an exact connection on my own just yet.
If you count oriented cycles, the number of such cycles for $n=2,3,4,\ldots$ is
$1, 6, 768, 3888000, \ldots$
If you count non-oriented cycles, ie edge-injective embeddings of the cyclic graph on $n(n-1)$ vertices onto $K_n$, the number of such cycles is
$1, 4, 384, 1944264, \ldots$
Neither sequence is in the OEIS, even omitting the first term, so it seems that such paths are likely not well studied (since no one seems to have even studied the enumeration problem before).
On the question of paths that visit each vertex as early as possible, the number of paths that start with $s_1\to s_2\ldots\to s_n$ is $1, 2, 36, 18432, \ldots$ for $n=2,3,4,5,\ldots$. For all such cycles, add a factor of $(n-1)!$