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How many tournament graphs are there up to Isomorphism?

Context: I am examining properties of complete directed graphs and so am trying to enumerate them on my simulation. I would like to know precisely how intractable this is.

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    $\begingroup$ oeis.org/A000568 $\endgroup$ – Mike Earnest Mar 21 at 23:15
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    $\begingroup$ Probably really complicated as Harary initially gave an incorrect calculation $\endgroup$ – Jorge Fernández Hidalgo Mar 21 at 23:20
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    $\begingroup$ @MikeEarnest No, A000568 is the number of tournaments (oriented complete graphs) of order $n$. As far as I know, there is only one complete digraph of order $n$, up to isomorphism. Each pair of vertices is joined by an arc in each direction, and there is a loop at each vertex. $\endgroup$ – bof Mar 22 at 5:50
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    $\begingroup$ Up to isomorphism, for each $n$ there is just one complete directed graph of order $n$. Are you sure it's complete directed graphs you're interested in, and not tournaments? $\endgroup$ – bof Mar 22 at 5:56

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