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Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n, listed by class size.
6

%I #13 Jan 27 2022 18:20:07

%S 0,0,2,0,2,0,4,6,2,0,0,4,5,0,46,73,2,0,0,0,4,0,0,8,0,2,36,0,43,2,473,

%T 1020,2,0,0,0,0,4,0,0,0,0,8,0,0,4,0,36,0,0,0,0,86,0,0,38,415,0,758,32,

%U 6682,18426,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,8

%N Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n, listed by class size.

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

%H Andrew Howroyd, <a href="/A079200/b079200.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>

%F A079194(n,k) + A079197(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).

%F A079198(n) = Sum_{k>=1} T(n,k)*A079210(n,k).

%F T(n,k) = A079175(n,k) - A079201(n,k). - _Andrew Howroyd_, Jan 26 2022

%e Triangle T(n,k) begins:

%e 0;

%e 0;

%e 2, 0;

%e 2, 0, 4, 6;

%e 2, 0, 0, 4, 5, 0, 46, 73;

%e 2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;

%e ...

%Y Row sums give A079199.

%Y Cf. A027423 (row lengths), A079171, A079194, A079175, A079197, A079198, A079199, A079201, A079210.

%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