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.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Robert Lauff, Nov 18 2022
STATUS
approved