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.

A079202 Number of isomorphism classes of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 32, 2155, 0, 0, 0, 12, 60, 184, 34544, 147032271
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,0; 0,0,32,2155; 0,0,0,12,60,184,34544,147032271
A079230(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079231(x).

Links

Crossrefs

Cf. A079203, A079204, A079205, A079197, A079207, A079208, A079209, A079230, A079231.

A079203 Number of isomorphism classes of non-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, 6, 34, 952, 0, 1, 12, 6, 181, 283, 13333, 31839187
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,2; 0,6,34,952; 0,1,12,6,181,283,13333,31839187
A079232(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079233(x).

Links

Crossrefs

Cf. A079202, A079204, A079205, A079197, A079207, A079208, A079209, A079232, A079233.

A079204 Number of isomorphism classes of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 146, 12992
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,0; 0,0,0,8; 0,0,0,0,0,0,146,12992
A079234(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079235(x).

Links

Crossrefs

Cf. A079202, A079203, A079205, A079197, A079207, A079208, A079209, A079234, A079235.

A079205 Number of isomorphism classes of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 29, 0, 237, 4374
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 2,0; 0,2,0,0; 0,0,2,0,29,0,237,4374
A079236(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079237(x).

Links

Crossrefs

Cf. A079202, A079203, A079204, A079197, A079207, A079208, A079209, A079236, A079237.

A079207 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 4, 4, 0, 46, 73, 0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 84, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8
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!)

Examples

			Triangle T(n,k) begins:
  0;
  0;
  0, 0;
  0, 0, 4, 6;
  0, 0, 0, 4, 4, 0, 46, 73;
  0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Links

Crossrefs

Row sums give A079241.
Cf. A027423 (row lengths), A079175, A079201, A079202, A079203, A079204, A079205, A079197, A079208, A079209, A079240.

Formula

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079208(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079240(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k) - A079208(n,k). - Andrew Howroyd, Jan 27 2022

Extensions

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 27 2022

A079209 Number of isomorphism classes of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 9, 0, 0, 0, 3, 0, 0, 16, 39
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,3; 0,0,3,9; 0,0,0,3,0,0,16,39
A079244(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079245(x).

Links

Crossrefs

Cf. A079202, A079203, A079204, A079205, A079197, A079207, A079208, A079244, A079245.

A079242 Number 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, 8, 2, 122, 2, 1682
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A079208, A079243 (isomorphism classes), A079230, A079232, A079234, A079236, A079195, A079240, A079244, A063524.

Formula

A079230(n) + A079232(n) + A079234(n) + A079236(n) + A079195(n) + A079240(n) + a(n) + A079244(n) + A063524(n) = A002489(n).
a(n) = Sum_{k>=1} A079208(n,k)*A079210(n,k).

Extensions

a(0)=0 prepended and terms 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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