A079241 -id:A079241 - cp's OEIS frontend

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.

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

A079231 Number of isomorphism classes 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, 2187, 147067071

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 A079202.

Cf. A079202, A079230, A079233, A079235, A079237, A079196, A079241, A079243, A079245, A063524.

A079233 Number of isomorphism classes 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, 2, 992, 31853003

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 A079203.

Cf. A079203, A079232, A079231, A079235, A079237, A079196, A079241, A079243, A079245, A063524.

A079235 Number of isomorphism classes 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, 8, 13138

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 A079204.

Cf. A079204, A079234, A079231, A079233, A079237, A079196, A079241, A079243, A079245, A063524.

A079237 Number of isomorphism classes 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, 2, 4642

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 A079205.

Cf. A079205, A079236, A079231, A079233, A079235, A079196, A079241, A079243, A079245, 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.

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

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

nonn

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

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

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

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

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

			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)

Index entries for sequences related to semigroups

Row sums of A079208.

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

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

a(0)=0 prepended and a(5)-a(8) from Andrew Howroyd, Jan 26 2022

cp's OEIS Frontend

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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A079231 Number of isomorphism classes of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

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A079233 Number of isomorphism classes of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

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A079235 Number of isomorphism classes of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n.

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A079237 Number of isomorphism classes of non-associative non-commutative anti-associative 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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A079240 Number of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

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

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