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Revision History for A350876

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Showing entries 1-10 | older changes
A350876
a(n) is the number of nonisomorphic flexible left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy all of the identities x(yx) = (xy)x, (xx)y = x(xy), and x(yy) = (xy)y (for all x and y).
(history; published version)
#14 by N. J. A. Sloane at Fri Feb 18 21:36:27 EST 2022
STATUS

proposed

approved

#13 by Andrew Howroyd at Tue Jan 25 22:08:35 EST 2022
STATUS

editing

proposed

#12 by Andrew Howroyd at Tue Jan 25 22:08:29 EST 2022
KEYWORD

nonn,hard,more,changed

STATUS

proposed

editing

#11 by Andrew Howroyd at Tue Jan 25 22:02:10 EST 2022
STATUS

editing

proposed

Discussion
Tue Jan 25
22:03
Andrew Howroyd: a(5) was 15 secs; a(6) 154 minutes; not even trying for a(7).
#10 by Andrew Howroyd at Tue Jan 25 21:55:09 EST 2022
DATA

1, 1, 5, 33, 688675, 65066, 41160471

CROSSREFS

Cf. A001329 (magmas), A350873 (flexible magmas), A350874 (left/right-alternative magmas), A350876 A350875 (left-right-alternative magmas).

EXTENSIONS

a(5)-a(6) from Andrew Howroyd, Jan 25 2022

STATUS

proposed

editing

Discussion
Tue Jan 25
22:02
Andrew Howroyd: Joel has conceded that the program he downloaded off twitter+github is buggy - and produces 13 extra magmas that don't comply with the given axioms. Since this matches the expected discrepancy, I think we can call this good (but the other 2 still have incorrect a(4)). Joel is also thinking he would like a limit of 7:)
#9 by Joel Brennan at Mon Jan 24 16:23:59 EST 2022
STATUS

editing

proposed

Discussion
Tue Jan 25
10:22
Andrew Howroyd: I sent you an email
#8 by Joel Brennan at Mon Jan 24 16:23:44 EST 2022
CROSSREFS

Cf. A001329 (magmas), A350873 (flexible magmas), A350874 (left/right-alternative magmas), A350876 (left-right-alternative magmas).

STATUS

proposed

editing

#7 by Robert C. Lyons at Sun Jan 23 20:16:22 EST 2022
STATUS

editing

proposed

Discussion
Sun Jan 23
21:43
Andrew Howroyd: Can you double check. I'm only finding 675? I'm also too low on the other sequences for the 4th term (especially A350874). For this one even if I filter the 72148 flexibles I only find 675.
21:44
Andrew Howroyd: I get the same values for n < 4 on all the sequences.
Mon Jan 24
07:55
Joel Brennan: I got these numbers using Andrej Bauer's program "alg" (see e.g. twitter.com/andrejbauer/status/1296555230184837122). I have double checked the ones which my laptop can compute in less than half an hour and I'm getting the same values. I will comment the .th files I used so you can try and replicate my values.
07:57
Joel Brennan: (Just an idea: perhaps you are looking at non-equivalent magmas instead of non-isomorphic magmas? - "equivalence" being "isomorphism or anti-isomorphism". This probably isn't the problem though as I would expect this to also give different values for n < 4)
07:59
Joel Brennan: The .th file I used for this sequence was "Binary *.
Axiom: y * (x * x) = (y * x) * x.
Axiom: x * (x * y) = (x * x) * y.
Axiom: x * (y * x) = (x * y) * x."
#6 by Robert C. Lyons at Sun Jan 23 20:16:04 EST 2022
COMMENTS

Compare A350873 and A350875, which are the numbers of flexible magmas with n elements and left-right-alternative magmas with n elements (up to isomorphism). The fact that a(n) < A350875(n) for n ≥ >= 3 means that left-right-alternativity for magmas (the identities (xx)y = x(xy) and x(yy) = (xy)y) does not imply flexibility (x(yx) = (xy)x). This is in contrast to the situation for non-associative rings, where this implication does hold (due to the additional additive structure).

STATUS

proposed

editing

#5 by Joel Brennan at Sun Jan 23 19:35:43 EST 2022
STATUS

editing

proposed

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