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 A242997 a(n) is the order of the group of invertible elements in the semigroup M whose elements are the closed binary operations on an n-point set S and whose operation (on operations, in this case) is given by x AB y = (x B y) A (y B x) for operations A and B on S and points x and y in S. 0

%I #23 Mar 31 2018 16:58:25

%S 1,4,288,1105920,445906944000,30851909057249280000,

%T 540013176648715369394995200000,

%U 3299903381977999900396941913809223680000000,9276369213749813701818662527515163802639831924736000000000

%N a(n) is the order of the group of invertible elements in the semigroup M whose elements are the closed binary operations on an n-point set S and whose operation (on operations, in this case) is given by x AB y = (x B y) A (y B x) for operations A and B on S and points x and y in S.

%C a(n) is also the number of permutations of the Cartesian square of an n-element set that commute with the permutation that sends each (x, y) to (y, x).

%C More generally, for the binary operation on k-ary operations on an n-set given by cyclically permuting k inputs to one operation to obtain k outputs to use as inputs to the other operation (as when k = 3, to illustrate, AB(x, y, z) = A(B(x, y, z), B(y, z, x), B(z, x, y))), the group of invertible operations is isomorphic to the centralizer of the cyclic permutation of coordinates, in the Symmetric Group on the k-th Cartesian power of the n-set, and the order of this group is Product(r divides k) Q(n, r)! r^Q(n, r) where Q(n, r) = (1/r) Sum_{d divides r} Mobius(r/d) n^d.

%D M. Hall, The Theory of Groups, MacMillan, 1959, 169-172.

%D N. Jacobson, Basic Algebra 1, 2nd Edition, W.H. Freeman, 1985, p. 289.

%H J. D. Reid, <a href="http://www.jstor.org/stable/2324878">On Finite Groups and Finite Fields</a>, The American Mathematical Monthly, Vol. 98, Num. 6, June-July 1991, pp. 549-551.

%F a(n) = n! * (n*(n-1)/2)! * 2^(n*(n-1)/2).

%e When n = 2, the 4 invertible binary operations are the left and right projections and the left and right "conjections", the left conjection being that which sends each (x, y) to "not x", which is unique when n = 2.

%o (PARI) a(n) = n! * (n*(n-1)/2)! * 2^(n*(n-1)/2);

%K nonn

%O 1,2

%A _David Pasino_, Aug 17 2014

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