I stuck with Logic, Computation and Set Theory by T. Forster.
In Ex. 9 p. 14 it is stated that on the given set the amount of antisymmetrical relations equals to the amount of trichotomous ones.
However I cannot get the same amount. E.g. lets take $2$-element set $\{a, b\}$. There are $12$ antisymmetrical relations (total number of antisymmetrical relations is $2^n3^{(n^2-n)/2}$ for $n$-element domain). I was able to count only three trichotomous ones, viz. $\{(a,b)\}$, $\{(b,a)\}$ and $\{(a,a), (b,b)\}$.
It's an error in the text; your count of antisymmetric relations is correct. You counted one trichotomous relation too many; $\{(a,a),(b,b)\}$ isn't trichotomous. There are $2^{n(n-1)/2}$ trichotomous relations, since we must have exactly one of $(x,y)$ for all $x\ne y$, and we mustn't have $(x,x)$ for any $x$.