A079209 -id:A079209 - cp's OEIS frontend

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.

0, 0, 0, 0, 0, 32, 2155, 0, 0, 0, 12, 60, 184, 34544, 147032271

Offset: 1

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

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).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

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

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

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).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

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

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

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).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

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

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

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).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

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

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

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums give A079241.

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

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

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

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

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

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

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums give A079243.

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

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).

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

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

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

nonn

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.

Index entries for sequences related to semigroups

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

A079245 Number of isomorphism classes of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 3, 12, 58

Offset: 1

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

nonn

A079231(n)+A079233(n)+A079235(n)+A079237(n)+A079196(n)+A079241(n)+A079243(n)+A079245(n)+A063524(n)=A002489(n)

Each a(n) is equal to the sum of the elements in row n of A079209.

Index entries for sequences related to semigroups

Cf. A079209, A079244, A079231, A079233, A079235, A079237, A079196, A079241, A079243, A063524.

cp's OEIS Frontend

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.

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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.

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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.

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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.

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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.

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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.

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

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A079245 Number of isomorphism classes of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

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