So if we have an ascending sequence of N, the numbers must all be in rank order, and the probability is 1/N!. Similarly if the longest ascending sequence is 1, the numbers must be in exact reverse order, so the probability is similarly 1/N!.
It follow that for a sequence of three the probabilities are 1/3!, 4/3! and 1/3!.
But how do I solve it for the general case, given a sequence of N, what the probability of a maximum ascending sequence of m?
(In case you're wondering, I need this for a gene synteny problem. Sequences of DNA match each other, and generally the order is preserved, but you also have some out of sequence matches, which might be false positives or translocations. So how many in-sequence matches do I need to declare significance?)