%I #37 Nov 04 2022 14:43:48

%S 2,14,544,173128,630988832,35355434970848

%N a(n) is the number of "generalized signotopes", 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 twice (equivalently +-+- and -+-+ are forbidden).

%C 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).

%C 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.

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

%D D. Knuth, Axioms and Hulls, Springer, 1992, 9-11.

%H M. Balko, R. Fulek, and J. Kynčl, <a href="http://doi.org/10.1007/s00454-014-9644-z">Crossing Numbers and Combinatorial Characterization of Monotone Drawings of K_n</a>, Discrete & Computational Geometry, Volume 53, Issue 1, 2015, Pages 107-143.

%H H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner, <a href="https://doi.org/10.1007/s00454-022-00408-6">Topological Drawings meet Classical Theorems from Convex Geometry</a>, Discrete & Computational Geometry, Springer, 2022.

%H S. Felsner and H. Weil, <a href="http://doi.org/10.1016/S0166-218X(00)00232-8">Sweeps, arrangements and signotopes</a>, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.

%H Manfred Scheucher, <a href="/A328377/a328377.cpp.txt">C-program for computing the first terms</a>

%Y Cf. A006245, A329980.

%K nonn,more,hard

%O 3,1

%A _Manfred Scheucher_, Oct 14 2019

%E a(8) from _Robert Lauff_ and _Manfred Scheucher_, Nov 04 2022