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%I #13 Jan 27 2022 15:41:58
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N 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.
%C Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
%C 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
%H Andrew Howroyd, <a href="/A079208/b079208.txt">Table of n, a(n) for n = 0..217</a> (rows 0..8)
%H 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>
%H <a href="/index/Se#semigroups">Index entries for sequences related to semigroups</a>
%F 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).
%F A079242(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).
%e Triangle T(n,k) begins:
%e 0;
%e 0;
%e 2, 0;
%e 2, 0, 0, 0;
%e 2, 0, 0, 0, 1, 0, 0, 0;
%e 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
%e ...
%Y Row sums give A079243.
%Y Cf. A027423 (row lengths), A079171, A079201, A079202, A079203, A079204, A079205, A079197, A079207, A079209, A079242.
%K nonn,tabf
%O 0,3
%A Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
%E a(0)=0 prepended and terms a(16) and beyond from _Andrew Howroyd_, Jan 26 2022
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