1. The OEIS strikes again! A conjectured closed form is now found from A124774 which is explained on the web page http://rangevoting.org/CombinedTestFail.html along with everything else I found out, which page I strongly suggest math-funners read, since they seem to keep not comprehending the problem, and maybe this page un-mysterifies. (I'm actually boggled at how much everybody cannot understand the problem, which proves it must be a great problem, and/or I am crazy.) 2. To answer James Propp
I do not understand Warren's example. Specifically, I do not know what he has in mind as the independent events whose conjunction is being considered.
Note that the event "there exists an ordering such that x_j<j/N for each j=1,2,3,...,N" is not the conjunction of the N events "there exists an ordering such that x_1<1/N", "there exists an ordering such that x_2<2/N", ..., "there exists an ordering such that x_N<N/N".
--No....: All N! orderings such as x1,x2,xN, and x2,x1,...,xN, and ... xN...x2,x1 are considered. What is the freaking chance, that at least one of these orderings obeys (after reordering) xj<j/N for each j=1,2,3,...,N? [Where each x_j is an iid uniform(0,1)]