The On-Line Encyclopedia of Integer Sequences (OEIS)

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A358471

a(n) is the number of transitive generalized signotopes.
(history; published version)

#23 by N. J. A. Sloane at Sun Dec 25 20:25:28 EST 2022

STATUS

proposed

approved

#22 by Sean A. Irvine at Sun Dec 25 17:47:56 EST 2022

STATUS

editing

proposed

#21 by Sean A. Irvine at Sun Dec 25 17:46:51 EST 2022

NAME

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

COMMENTS

A "transitive generalized signotope" is a generalized signotopes 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's Book.

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

CROSSREFS

Cf. A006247, A328377.

STATUS

proposed

editing

#20 by Robert Lauff at Thu Nov 24 08:10:50 EST 2022

STATUS

editing

proposed

Discussion

Thu Nov 24

08:14

Robert Lauff: @Jon E. Schoenfield.
The capital "Signotope" was a typo, thank you for catching that.
Transitive generalized signotopes are special generalized signotopes. And "normal" signotopes are also special generalized signotopes (in signotopes, only one sign switch in X(abcd) is allowed. In generalized signotopes we allow two switches). The statement then is:
signotopes < transitive generalized signotopes < generalized signotopes,
where < is the subset relation.

11:54

Jon E. Schoenfield: Okay, thanks!

#19 by Robert Lauff at Thu Nov 24 08:10:40 EST 2022

COMMENTS

Every Signotope 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.

STATUS

proposed

editing

#18 by Jon E. Schoenfield at Mon Nov 21 20:07:49 EST 2022

STATUS

editing

proposed

Discussion

Thu Nov 24

02:31

Jon E. Schoenfield: Why is "Signotope" capitalized?

02:34

Jon E. Schoenfield: Does the statement that "Every Signotope ... is a transitive generalized signotope ..." mean that there's no such thing as a signotope that is not a transitive generalized signotope?  The wording "transitive generalized signotope" seems to me to at least imply that not all signotopes are transitive generalized.... (I'm sorry, I don't know much about the subject matter for this sequence, but the wording seems strange to me.)

#17 by Jon E. Schoenfield at Mon Nov 21 20:07:47 EST 2022

COMMENTS

Every Signotope (cf. A006247) is a transitive generalized signotope, giving a lower bound of 2^(cnc*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.

STATUS

proposed

editing

#16 by Robert Lauff at Sun Nov 20 04:52:05 EST 2022

STATUS

editing

proposed

#15 by Robert Lauff at Sun Nov 20 04:51:57 EST 2022

COMMENTS

Every Signotope (cf. A006247) is a transitive generalized signotope, giving a lower bound of 2^(cn^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.

STATUS

proposed

editing

#14 by Robert Lauff at Sat Nov 19 19:37:39 EST 2022

STATUS

editing

proposed