Of course it is a special case of the question: Given a binary operation on a set X determine the probability that two elements of X chosen uniformly at random commute. This question has been studied for groups and semigroups. In particular this paper http://archive.maths.nuim.ie/staff/sbuckley/Papers/semigp_cp.pdf finds the order Ord(i,j) of the smallest semigroup whose commuting probability is i/j. There is a Mathoverflow discussion of your particular problem at http://mathoverflow.net/questions/143058/what-is-the-probability-two-random-... On Mon, Jul 21, 2014 at 1:37 PM, Joerg Arndt <arndt@jjj.de> wrote:
Hope it is OK to occasionally(!) raise an OEIS related question. This is about A181162 ( https://oeis.org/draft/A181162 ).
I have no better idea than the author: exhaustive search. It seems almost certain that this (nice) problem has been investigated somewhere. Any pointers?
Best, jj
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