Descending subsequences of random permutations

Given a random permutation of the numbers 1, 2, . . . . n, let L, be the length of the longest descending subsequence of this permutation. Let F, be the minimal header (first element) of the descending subsequences having maximal length. It is known that EL,/&,,,, c and that c=2. However, the proofs that r=2 are far from elementary and involve limit processes. Several relationships between these two random variables are established, namely, EL, = ET=, P(F, = j) and P( F, + , =n+l)=l-EF,/n+l. Some other combinatorial identities regarding the distribution of the bivariate random variable (L,, F,,) are also proved. The definition of F, is generalized, characterizing the elements appearing at the first row and first column of the Young tableau corresponding to a given permutation. As a result. an elementary proof for c<2 is constructed.