The number of possible commutative binary operations that can be defined on a set of n elements (for a given n) is _____...

Question

The number of possible commutative binary operations that can be defined on a set of n elements (for a given n) is ___________.

Accepted Answer

n^(n(n+1)/2) commutative binary operations A binary operation on a set S of n elements is just a function that takes an ordered pair (a, b) and returns one of the n elements - so it is fully described by filling an n x n operation table. Commutativity means a*b = b*a, so the table is symmetric across its main diagonal; once you fill the diagonal and the upper triangle, the lower triangle is forced. So you only freely choose values for the n diagonal cells plus the C(n,2) cells above it, and each of those n(n+1)/2 cells can independently be any of the n elements.