OEIS has
* [A001403](https://oeis.org/A001403) Number of combinatorial configurations of type (n_3).
* [A100001](https://oeis.org/A100001) Number of self-dual combinatorial configurations of type (n_3).
They first differ at $n=11$. An example of a non-self-dual configuration of type $(11_3)$ has points $0$ to $10$ and lines $[0, 1, 2]$, $[0, 3, 4]$, $[0, 5, 6]$, $[1, 3, 7]$, $[1, 5, 8]$, $[4, 5, 9]$, $[3, 8, 10]$, $[6, 8, 9]$, $[2, 9, 10]$, $[6, 7, 10]$, $[2, 4, 7]$.
This was constructed by using nauty to generate regular bipartite graphs, filtering to configurations, and testing the automorphism groups. (And, for what it's worth, I found the OEIS entries on the basis of counts for small $n$ produced during the search).
Peter Taylor
·
2024-01-05T00:33:56Z (11 months ago)