Let $r:A\to\mathbb{N}$ be a rank mapping, surjective on $\{1,..,k\}\subset \mathbb{N}$
The mapping $r$ defines a relation $\sigma$: $$x\sigma y\iff r(x)<r(y)$$
The set of all possible relations on $A$ to be found through such a rank mapping, is then $$P_A = \{\sigma \subset A \times A | \sigma \text{ is generated by a rank mapping } r\}$$
Combined with this we also obtain $x\sigma^* y \iff r(x)\leq r(y)$ with the associated set $P_A^*$ of such relations.
My question is how to count the number of preorders $n$ on the finite set $A=\{a,b,c\}$. This corresponds to the cardinality of $P_A^*$. According to OEIS, the result should be $n=19$.
In order to produce the answer, I partitioned the possible relations in $P_A*$ into 3 groups. The ones which map onto 1 element, 2 elements and 3 elements. This results into 1 (a, b and c map to rank 1), 6 (a,b and c map to rank 1 or 2) and 6 (a, b and c map to rank 1, 2 or 3), which gives 13.
How can I tackle this problem?