Not really; at least none that I know of. My last paragraph was really an answer to this question. "There really isn't any obvious choice for the distribution." I don't think measure theory will help. I don't know of any interesting measures on functions from integers to integers; even if we did choose one, the permutations would almost have to be measure zero. A measure on the permutations seems even less likely. One can build a sequence of distributions based on a parameterized distribution. Let P(N,x) be a distribution of positive integers that become "flat" as N->infinity. The x here is a random variable on the interval (0,1). One could, for example, take P(N,x) = floor(-N*log(x)). Choose a parameter n. Start with p completely undefined. For each k from 1 to infinity, if p(k) is not yet defined, choose a random number x_k, and define p(k) = b_(P(n*k,x_k)), where b is the list of all numbers not yet in the range of the p. Then, if there is as yet no m with p(m) = k, choose a random number y_k, take m = c_(P(n*k,y_k)), where c is the list of numbers for which p is not yet defined, and set p(m) = k. Now take the probability that this permutation is a derangement, and take the limit as n->infinity. Note that there are still two arbitrary things about this procedure. The choice of P is obviously one; but it isn't clear that we should use P(n*k,...) instead of, perhaps, P(n+k,...) or P(n^k,...) or whatever. (We can't use P(n,...), because then there would be the same finite chance that 1 is selected at each step, and thus the probability of a derangement would be 0.) It might be that, under suitable assumptions about P, you could show that this limit is 1/e. Even if you did, I don't know how much it would really tell you. Franklin T. Adams-Watters -----Original Message----- From: wrcordw@sandia.gov Well, that might be true, but is there any sense (maybe measure theory?), even if I don't have a nicely defined concept of picking a value at random, where I can still talk about the relative probability? --Bill C. -----Original Message----- From:franktaw@netscape.net Picking an infinite permutation at random is not a well defined concept. It comes down to the similar but simpler problem of picking a (positive) integer at random. Each integer must have probability zero (lim_{n->infinity} 1/n} of being chosen; but that means you can't ever pick one. For a random infinite permutation p, p(1) needs to be a positive integer picked at random. --- If you choose a probability distribution for infinite permutations, the probability that one will be a derangement will depend on what probability distribution you choose. There really isn't any obvious choice for the distribution. Franklin T. Adams-Watters -----Original Message----- From: wrcordw@sandia.gov For n objects, the fraction of permutations that are derangements is about 1/e. Does this extend to an infinite number of objects; i.e. is the probability of picking a permutation "at random" that is a derangement = 1/e? --Bill C. ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.