[math-fun] probability random permutation fixes at least one odd-cardinality prefix

The following problem recently arose (and was only approximately solved) in a paper in Notices AMS. Can mathfunners do better? GIVEN: a random permutation of {1,2,3,...,n} where n>2. QUESTION: what is the probability P(n) that the permutation fixes at least one odd-cardinality prefix? For example, if the prefix length is k, and k=odd, then this probability would be k! * (n-k)! / n!. However, the problem is asking not about just one prefix-length k, but ALL odd k: k=1,3,5,.. whichever is odd among {n-1, n-2}. The Notices paper claimed P(n) <= 2/n + O(n^(-2)). Can one say something more precise? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)