Two eulerian trails of $K_{2n+1}$ are defined to be equivalent if the orientations obtained by orienting the edges as traversed by the trails are isomorphic as digraph. How many nonequivalent trails are there?
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$\begingroup$ The eulerian trails $0\to 1\to 2\to 0$ and $0\to 2\to 1\to 0$ of the complete graph $K_3$ are considered nonequivalent? $\endgroup$ – Freddy Barrera Apr 22 at 12:28

2$\begingroup$ @FreddyBarrera Those would be considered equivalent since they are isomorphic as digraphs (both are just directed triangles). $\endgroup$ – Tony Huynh Apr 22 at 13:14

$\begingroup$ @FreddyBarrera, yes they are the same. Number for $K_3$ is 1. $\endgroup$ – hbm Apr 22 at 13:49
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If I understand your question, you are asking for the number of isomorphism classes of regular tournaments.
There are no exact formulas. See http://oeis.org/A096368 for counts up to 15 vertices.
The asymptotic number is known. It is $RT(2n+1)/(2n+1)!$, where $RT(2n+1)$ is given in the abstract of this paper.