A079242 -id:A079242

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.

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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

LINKS

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

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).

EXAMPLE

Triangle T(n,k) begins:

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;

...

CROSSREFS

Row sums give A079243.

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

KEYWORD

nonn,tabf

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A079230

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

+10
9

0, 0, 13026, 3529190912

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..4.

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A079232

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

+10
9

0, 4, 5826, 764303896

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..4.

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A079234

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

+10
9

0, 0, 48, 313560

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..4.

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A079236

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

+10
9

0, 2, 4, 108000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..4.

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A079244

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

+10
9

0, 6, 63, 1140

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..4.

Index entries for sequences related to semigroups

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A079240

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..8.

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A079243

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

+10
8

0, 0, 2, 2, 3, 2, 4, 2, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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

LINKS

Table of n, a(n) for n=0..8.

Index entries for sequences related to semigroups

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

EXAMPLE

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)

CROSSREFS

Row sums of A079208.

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

KEYWORD

nonn,more

AUTHOR

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

EXTENSIONS

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

STATUS

approved