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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.
+10
9
0, 0, 0, 0, 0, 32, 2155, 0, 0, 0, 12, 60, 184, 34544, 147032271
COMMENTS
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).
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
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.
+10
9
0, 0, 2, 0, 6, 34, 952, 0, 1, 12, 6, 181, 283, 13333, 31839187
COMMENTS
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).
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
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.
+10
9
0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 146, 12992
COMMENTS
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).
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
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.
+10
9
0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 29, 0, 237, 4374
COMMENTS
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).
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
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.
+10
9
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
COMMENTS
Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
EXAMPLE
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;
...
CROSSREFS
Cf. A027423 (row lengths), A079175, A079201, A079202, A079203, A079204, A079205, A079197, A079208, A079209, A079240.
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 27 2022
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.
+10
9
0, 0, 3, 0, 0, 3, 9, 0, 0, 0, 3, 0, 0, 16, 39
COMMENTS
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).
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
Number of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.
+10
8
0, 0, 2, 2, 8, 2, 122, 2, 1682
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
EXTENSIONS
a(0)=0 prepended and terms a(5)-a(8) added by Andrew Howroyd, Jan 27 2022
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
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
EXAMPLE
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)
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
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