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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

C. van den Bosch, <a href="https://web.archive.org/web/20071014230143/http://cosmos.ucc.ie/~cjvdb1/html/binops.shtml">Closed binary operations on small sets</a>

Andrew Howroyd, <a href="/A079208/b079208.txt">Table of n, a(n) for n = 0..217</a> (rows 0..8)

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079207(n) + a(n,k) + A079209T(n,k) + A063524A079201(n,k) = A079171(n,k).

Cf. A027423 (row lengths), A079171, A079201, A079202, A079203, A079204, A079205, A079197, A079207, A079209, A079242, A079243.

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

The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The only 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

A079202(n) + A079203(n) + A079204(n) + A079205(n) + A079197(n) + A079207(n) + A079208a(n) + A079209(n) + A063524(n) = A079171(n).

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

1,2

0,3

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)

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

First four rows: 0; 2,0; 2,0,0,0; 2,0,0,0,1,0,0,0

A079242(x) is equal to the sum The only closed binary operations that are both commutative and anti-commutative are those on sets of the products size <= 1. The only significance of each element non-commutative and non-anti-associative in row x of the name is that it excludes this sequence and possibility. Otherwise, the corresponding element of A079210first two terms would be 1. - _Andrew Howroyd_, Jan 26 2022

The sum of each row x of this sequence is given by A079243(x).

A079202(n) + A079203(n) + A079204(n) + A079205(n) + A079197(n) + A079207(n) + A079208(n) + A079209(n) + A063524(n) = A079171(n).

A079242(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

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;

...

Row sums give A079243.

Cf. A027423 (row lengths), A079202, A079203, A079204, A079205, A079197, A079207, A079209, A079242, A079243.

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

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<a href="/Sindx_index/Se.html#semigroups">Index entries for sequences related to semigroups</a>

<a href="/Sindx_Se.html#semigroups">Index entries for sequences related to semigroups</a>

nonn,tabf,new

<a href="http://www.research.att.com/~njas/sequences/Sindx_Se.html#semigroups">Index entries for sequences related to semigroups</a>

nonn,tabf,new