TITLE: Non-equivalent eulerian trails in $K_{2n+1}$
QUESTION [6 upvotes]: 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 non-equivalent trails are there?

REPLY [4 votes]: 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.