Did I ever screw up that answer by not reading the question. Still: Neil -- if you're out there -- the disordered hats problem suggests two new sequences which a search on the word "derangement" did not find in the OEIS: a) the numerators, and b) the denominators of the (always rational) probability (in lowest terms) that a random permutation of n items is a derangement. Q: How often are the numerators prime? --Dan the red-faced