Yes, but most (in some sense) permutations are infinitely far apart by this metric. Or, to put it another way, if a given infinite permutation has infinitely many fixed points, every permutation in any neighborhood of it has infinitely many fixed points, and thus is not a derangement. Further, there are infinitely many permutations one transposition away from a given infinite permutation, and trying to pick one at random gets us right back to our problem of picking a random integer. (We can get around this problem by making the distance be the maximum value at which the permutations differ, but this doesn't help with the first problem.) Fundamentally, this approach won't work. It requires that the neighborhood be finite, but that every permutation be in some neighborhood. This is inconsistent with the fact that there are uncountably many permutations. (You could instead ask for a measure on each neighborhood, but then you are really getting a measure on the set of all infinite permutations. If you have that, you don't need the neighborhoods.) Franklin T. Adams-Watters -----Original Message----- From: mike@math.ucr.edu You can talk about the density of primes near an integer, something that is often called "the probability that a randomly chosen number is prime" even though that statement doesn't parse. You could therefore talk about the density of derangements near a given permutation, where distance between permutations is the number of transpositions required to transform one into the other. -- Mike Stay ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.