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.

1, 4, 288, 1105920, 445906944000, 30851909057249280000, 540013176648715369394995200000, 3299903381977999900396941913809223680000000, 9276369213749813701818662527515163802639831924736000000000

Offset: 1

Author

David Pasino, Aug 17 2014

Keywords

• nonn

Comments

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).

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.

Examples

			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.
		

References

• M. Hall, The Theory of Groups, MacMillan, 1959, 169-172.
• N. Jacobson, Basic Algebra 1, 2nd Edition, W.H. Freeman, 1985, p. 289.

Links

• J. D. Reid, On Finite Groups and Finite Fields, The American Mathematical Monthly, Vol. 98, Num. 6, June-July 1991, pp. 549-551.

Programs

PARI

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

Formula

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