Number of vacuously transitive relations A relation $R$ on a set $A$ is vacuously transitive if it is transitive but there do not exist ordered pairs such as $(x,y)$,$(y,z)$,$(x,z)$ .
If $A$ has $n$ elements, what is the number of such relations on $A$?
I tried for some small sets. For the null set, it is $1$. For the singleton, it is $2$.
OEIS lists a few more terms here. It shows the value $7$ for $n=2$. I am not getting $7$ though.
 A: The OEIS entry that you mention has the reference

H. Sharp, Jr., Enumeration of vacuously transitive relations, Discrete
Math. 4 (1973), 185-196.

If you follow that up, you find an explicit listing of the relations for $n=2$ and $3$ on page 194. For $n=2$ they are, in matrix notation:
$$
\begin{bmatrix}0&0 \\ 0&0\end{bmatrix},\begin{bmatrix}1&0 \\ 0&0\end{bmatrix},\begin{bmatrix}1&0 \\ 0&1\end{bmatrix},\\
\begin{bmatrix}0&0 \\ 1&0\end{bmatrix},\begin{bmatrix}1&0 \\ 1&0\end{bmatrix},\begin{bmatrix}1&0 \\ 1&1\end{bmatrix},\begin{bmatrix}0&0 \\ 1&1\end{bmatrix}.
$$
so check which one(s) you may have missed.
About the definition of vacuously transitive
EDIT (after comments). Sharp's definition is a bit vague. For completeness here is his definition:

Definition 2.1. A transitive triple in a relation is a set of three ordered pairs $\{(s_i,s_j),(s_j,s_k),(s_i,s_k)\}$. A relation is
called vacuously transitive if it is transitive but contains no
transitive triple.

At face value this seems to disallow self-loops in the graph altogether: for if we have $(a,a)$ in the relation, isn't $T=\{(a,a),(a,a),(a,a)\}$ a transitive triple? Yet Sharp clearly allows self-loops.
After reading the definition several times, I think the key is "a set of three ordered pairs". The $T$ above is a set of only one ordered pair, so it is not a transitive triple. So loops per se are allowed in a vacuously transitive relation.
However, in the 2-element relation $\{(1,1),(1,2),(2,1)\}$ we do have a transitive triple (the full relation is such a triple!) so this is not a vacuously transitive relation.
I have to say the definition could have been clearer.
About isomorphism
Another thing to note is that Sharp is explicitly counting isomorphism classes of the relations (i.e. up to permutation of elements). For example, the second of his relations above is $\{(1,1)\}$. He is not counting $\{(2,2)\}$ separately, because it is isomorphic to the previous (the isomorphism is, of course, $f(1)=2, f(2)=1$).
The OEIS entry A003041 does not explicitly say "isomorphism", but since the numbers are from Sharp's paper and that is the only source cited (besides the printed Encyclopedia), we can deduce that the OEIS entry is meant to be about isomorphism classes. I will try to refine the entry. (UPDATE: The OEIS entry is now better.)
