cp's OEIS Frontend

Clearly a generalization of "signotopes" (cf. A006245), i.e., mappings X:{{1..n} choose 3}->{+,-} such that for any four indices a < b < c < d, the sequence X(a,b,c), X(a,b,d), X(a,c,d), X(b,c,d) changes its sign at most once (see Felsner-Weil and Balko-Fulek-Kynčl reference).

Also a generalization of "simple topological drawings" (a.k.a. good drawings, cf. A276109), i.e., non-isomorphic drawings of the complete graph K_n such that any two edges intersect at most once. In a simple topological drawings, each three vertices a < b < c determine a triangle which is either oriented clockwise or counterclockwise -- this clearly motivates the mapping X. It can be checked that in any simple topological drawing of K_4, the sequence X(a,b,c), X(a,b,d), X(a,c,d), X(b,c,d) changes its sign at most twice.

Also known as "Interior triple systems", see Knuth's book.

The number of realizable order types on n points in the plane (A063666) is exactly the number of rotation systems of straight-line drawings of K_n.