Enumeration of trichotomous relations 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 $x\
e y$, and we mustn't have $(x,x)$ for any $x$.