|
%I #8 Mar 30 2012 18:40:58
%S 0,4,23,186,1914,28632,1627671,3684030412,105978177936290
%N Number of nonisomorphic semigroups of order n minus number of groups of order n.
%C In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Semigroup.html">Semigroup</a>
%F a(n) = A027851(n) - A000001(n).
%e a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1 = 0.
%e a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1 = 4.
%Y Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116.
%K nonn,hard,less
%O 1,2
%A _Jonathan Vos Post_, Feb 13 2011
|
