A358471 a(n) is the number of transitive generalized signotopes.

2, 14, 424, 58264, 33398288, 68779723376

Offset: 3

Author

Robert Lauff, Nov 18 2022

Keywords

• nonn
• more

Comments

A "transitive generalized signotope" is a generalized signotope X (cf. A328377) with the additional property that for any 5-tuple p, q, r, s, t, if (X(t,q,r), X(p,t,r), X(p,q,t), X(s,q,t), X(p,s,t), X(p,q,s)) = (+,+,+,+,+,+), then X(s,q,r)=+. Here X is extended to non-ordered triples by X(p(a),p(b),p(c)) = sgn(p)X(a,b,c) for any permutation p of three elements.

The "transitivity property" from the definition has a nice interpretation in the context of point sets, see "transitive interior triple systems" in Knuth.

The condition of transitivity from the definition above is implication (2.4a) in Knuth.

Every signotope (cf. A006247) is a transitive generalized signotope, giving a lower bound of 2^(c*n^2) <= a(n). This can be seen by checking the n=5 case. A violating 5-tuple in any signotope then cannot occur because it induces a signotope on 5 elements.

References

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

Crossrefs

Cf. A006247, A328377.