cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A350873 a(n) is the number of non-isomorphic flexible magmas with n elements (i.e., number of magmas with n elements which satisfy the identity (xy)x = x(yx) for all x and y).

Original entry on oeis.org

1, 1, 6, 191, 72148
Offset: 0

Views

Author

Joel Brennan, Jan 20 2022

Keywords

Examples

			There are 10 non-isomorphic magmas with 2 elements, and 6 of these are flexible magmas, so a(2) = 6.

Links

Crossrefs

Cf. A001329 (magmas), A027851.

A350874 a(n) is the number of nonisomorphic magmas with n elements which satisfy the identity (xx)y = x(xy) for all x and y (so-called left-alternative magmas).

Original entry on oeis.org

1, 1, 5, 97, 25311, 165974649
Offset: 0

Views

Author

Joel Brennan, Jan 20 2022

Keywords

Comments

Equivalently (by symmetry), a(n) also equals the number of nonisomorphic right-alternative magmas with n elements (that is, magmas satisfying the identity x(yy) = (xy)y for all x and y).

Examples

			There are 10 non-isomorphic magmas with 2 elements, and 5 of these are left-alternative, so a(2) = 5.

Crossrefs

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

Extensions

a(5) from Andrew Howroyd, Jan 29 2022

A350875 a(n) is the number of nonisomorphic left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy the identities (xx)y = x(xy) and x(yy) = (xy)y for all x and y.

Original entry on oeis.org

1, 1, 5, 43, 2027, 1005972
Offset: 0

Views

Author

Joel Brennan, Jan 20 2022

Keywords

Comments

Compare with A350876, whose terms are smaller (for n > 2) - this means that the left and right alternative identities (xx)y = x(xy) and x(yy) = (xy)y do not imply the flexible identity (xy)x = x(yx) for magmas. This is in contrast to the situation for non-associative rings, where left-right-alternativity implies flexibility (due to the additional additive structure).
a(n) = A350874(n) for n <= 2, i.e., a magma with (zero, one or) two elements which is left (resp., right) alternative is also right (resp., left) alternative.

Examples

			There are 10 nonisomorphic magmas with 2 elements, 5 of which are left-right-alternative, so a(2) = 5.
Similarly there are 3330 nonisomorphic magmas with 3 elements, 43 of which are left-right-alternative, so a(3) = 43.

Crossrefs

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

Extensions

a(5) from Andrew Howroyd, Jan 29 2022

A362642 Number of nonisomorphic magmas with n elements satisfying the equations (xy)y = x and x(yz) = xy.

Original entry on oeis.org

1, 1, 2, 4, 13, 47, 255, 1810, 18471, 266931
Offset: 0

Views

Author

Andrew Howroyd, Apr 28 2023

Keywords

Crossrefs

Cf. A001329 (magmas), A361720, A362382, A362385, A362643 (labeled case).

A362822 Number of nonisomorphic magmas with n elements satisfying the identities (xy)y = x and (xy)z = (xz)y.

Original entry on oeis.org

1, 1, 3, 6, 68, 254, 14310, 112762, 43419892
Offset: 0

Views

Author

Andrew Howroyd, May 06 2023

Keywords

Crossrefs

Cf. A001329 (magmas), A361720, A362382, A362642, A362823 (labeled case).

A006448 Number of n-element algebras with 1 binary operator and 1 constant (pointed groupoids).

Original entry on oeis.org

1, 16, 9882, 715860992, 12417636281312500, 85953540009068492207916672, 356838302112667713247240882121025536249, 1245456693529103515171728481423145699858332531028201472
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001329.

Formula

For a list n(1), n(2), n(3), ..., let fixF[n] = n(1)*Product_{i, j >= 1} (Sum_{d|[ i, j ]} (d*n(d))^((i, j)*n(i)*n(j))), where [i,j] = lcm(i,j). [Note that the notation fixF[n] appears in several other formulas contributed by Christian G. Bower. In this case it seems that the prefix "fix" was accidentally removed over the course of the years. - N. J. A. Sloane, Dec 19 2021]
a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = s_1 * Product_{i, j>=1} ( (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)).
a(n) is asymptotic to n^(n^2+1)/n!.

Extensions

Formula and more terms from Christian G. Bower, May 08 1998, Dec 03 2003

A030250 Number of nonisomorphic groupoids with no idempotents.

Original entry on oeis.org

1, 0, 3, 978, 56630832, 813802235250650, 4797611106478930201025520, 17327898558894838031017212608777367670, 53493753018329066207577438898229132113128344235601408
Offset: 0

Views

Author

Christian G. Bower, Feb 15 1998 and May 15 1998

Keywords

Links

Crossrefs

Cf. A001329, A038018.

A030260 Number of nonisomorphic commutative groupoids with no idempotents.

Original entry on oeis.org

1, 0, 1, 38, 13872, 83360520, 10203847031340, 31023254154131753920, 2765562268014305034000397632, 8332535835277886736134596954072281240, 960864308045670310058158724983067048253497223280
Offset: 0

Views

Author

Christian G. Bower, Feb 15 1998 and May 15 1998

Keywords

Links

Crossrefs

Cf. A001329, A001425, A038021.

A038017 Number of n-element commutative groupoids with an identity ("pointed" groupoids).

Original entry on oeis.org

1, 2, 15, 720, 409600, 3920030472, 775775333825891, 3837862827737186253664, 558740081065710564284870598075, 2755731923933734753149997221152548428020, 520996314135332606285488148844494695722050333912483
Offset: 1

Views

Author

Christian G. Bower, May 15 1998; revised Dec 05 2003

Keywords

Comments

Also number of commutative partial groupoids with n-1 elements or commutative groupoids with an absorbant (zero) element with n elements.

Links

Crossrefs

Cf. A001329, A030257.

Formula

a(n+1) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (1 + sum {d|i} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (1 + sum {d|i} (d*s_d))^(i*s_i^2/2) * (1 + sum {d|i/2} (d*s_d))^s_i or {i != j} (1 + sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)
a(n) asymptotic to (n^binomial(n, 2)+1)/n! = A090599(n)/A000142(n) = A076113(n)/A000142(n-1)

A038023 Triangle: T(n,k), k<=n: commutative groupoids with a nontrivial symmetry with n elements and k idempotents.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 6, 0, 4, 96, 214, 128, 64, 38, 54248, 83925, 54223, 28254, 10223, 1894
Offset: 0

Views

Author

Christian G. Bower, May 15 1998

Keywords

Links

Crossrefs

Cf. A001329, A001425, A030259, A030262, A030265.

Formula

Difference of A038021 and A038022.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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