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

2, 14, 424, 58264, 33398288, 68779723376

Offset

3,1

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.

Links

Table of n, a(n) for n=3..8.

Crossrefs

Cf. A006247, A328377.

Sequence in context: A135865 A337073 A306815 * A355872 A307123 A324306

Adjacent sequences: A358468 A358469 A358470 * A358472 A358473 A358474

Keyword

nonn,more

Author

Robert Lauff, Nov 18 2022

Status

approved