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Search: a079241 -id:a079241
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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
OFFSET
0,7
COMMENTS
Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
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
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
Row sums give A079241.
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 27 2022
STATUS
approved
Number of isomorphism classes of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.
+10
9
0, 0, 2187, 147067071
OFFSET
1,3
COMMENTS
Each a(n) is equal to the sum of the elements in row n of A079202.
KEYWORD
nonn
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
STATUS
approved
Number of isomorphism classes of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.
+10
9
0, 2, 992, 31853003
OFFSET
1,2
COMMENTS
Each a(n) is equal to the sum of the elements in row n of A079203.
KEYWORD
nonn
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
STATUS
approved
Number of isomorphism classes of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n.
+10
9
0, 0, 8, 13138
OFFSET
1,3
COMMENTS
Each a(n) is equal to the sum of the elements in row n of A079204.
KEYWORD
nonn
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
STATUS
approved
Number of isomorphism classes of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n.
+10
9
0, 2, 2, 4642
OFFSET
1,2
COMMENTS
Each a(n) is equal to the sum of the elements in row n of A079205.
KEYWORD
nonn
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
STATUS
approved
Number of isomorphism classes of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.
+10
9
OFFSET
1,2
COMMENTS
Each a(n) is equal to the sum of the elements in row n of A079209.
KEYWORD
nonn
AUTHOR
Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
STATUS
approved
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
OFFSET
0,4
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
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
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
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
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.
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

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