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.

Showing 1-8 of 8 results.

A079208 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The significance of non-commutative (and non-anti-associative) in the name is that it excludes this possibility. Otherwise, the first two terms would be 1. - Andrew Howroyd, Jan 26 2022

Examples

			Triangle T(n,k) begins:
  0;
  0;
  2, 0;
  2, 0, 0, 0;
  2, 0, 0, 0, 1, 0, 0, 0;
  2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Links

Crossrefs

Row sums give A079243.
Cf. A027423 (row lengths), A079171, A079201, A079202, A079203, A079204, A079205, A079197, A079207, A079209, A079242.

Formula

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079207(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079242(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

Extensions

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079230 Number of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 13026, 3529190912
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079202 and the corresponding element of A079210.

Crossrefs

Cf. A079202, A079231, A079232, A079234, A079236, A079195, A079240, A079242, A079244, A063524.

A079232 Number of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 4, 5826, 764303896
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079203 and the corresponding element of A079210.

Crossrefs

Cf. A079203, A079233, A079230, A079234, A079236, A079195, A079240, A079242, A079244, A063524.

A079234 Number of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 48, 313560
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079204 and the corresponding element of A079210.

Crossrefs

Cf. A079204, A079235, A079230, A079232, A079236, A079195, A079240, A079242, A079244, A063524.

A079236 Number of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 4, 108000
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079205 and the corresponding element of A079210.

Crossrefs

Cf. A079205, A079237, A079230, A079232, A079234, A079195, A079240, A079242, A079244, A063524.

A079244 Number of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 6, 63, 1140
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079209 and the corresponding element of A079210.

Links

Crossrefs

Cf. A079209, A079245, A079230, A079232, A079234, A079236, A079195, A079240, A079242, A063524.

A079240 Number of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 0, 48, 2344, 153000, 15875924, 7676692856, 148188196673360
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Crossrefs

Cf. A079207, A079241 (isomorphism classes), A079230, A079232, A079234, A079236, A079195, A079242, A079244, A063524.

Formula

A079230(n) + A079232(n) + A079234(n) + A079236(n) + A079195(n) + a(n) + A079242(n) + A079244(n) + A063524(n) = A002489(n).
a(n) = Sum_{k>=1} A079207(n,k)*A079210(n,k).
a(n) = A023814(n) - A023815(n) - A079242(n). - Andrew Howroyd, Jan 27 2022

Extensions

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 27 2022

A079243 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 2, 2, 3, 2, 4, 2, 4
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The significance of non-commutative (and non-anti-associative) in the name is that it excludes this possibility. Otherwise, the first two terms would be 1. - Andrew Howroyd, Jan 26 2022

Examples

			From _Andrew Howroyd_, Jan 26 2022: (Start)
The a(6) = 4 operations are the two shown below and their converses.
    | 1 2 3 4 5 6         | 1 2 3 4 5 6
  --+------------       --+------------
  1 | 1 2 3 4 5 6       1 | 1 2 3 1 2 3
  2 | 1 2 3 4 5 6       2 | 1 2 3 1 2 3
  3 | 1 2 3 4 5 6       3 | 1 2 3 1 2 3
  4 | 1 2 3 4 5 6       4 | 4 5 6 4 5 6
  5 | 1 2 3 4 5 6       5 | 4 5 6 4 5 6
  6 | 1 2 3 4 5 6       6 | 4 5 6 4 5 6
(End)

Links

Crossrefs

Row sums of A079208.
Cf. A079242 (labeled case), A079231, A079233, A079235, A079237, A079196, A079241, A079245, A063524.

Formula

A079231(n) + A079233(n) + A079235(n) + A079237(n) + A079196(n) + A079241(n) + a(n) + A079245(n) + A063524(n) = A002489(n).
Conjecture: a(n) = A000005(n) for n > 1. - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and a(5)-a(8) from Andrew Howroyd, Jan 26 2022
Showing 1-8 of 8 results.

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