On the longest upsequence problem for permutations

Given a permutation of n numbers, its longest upsequence can be found in time O(nlog logn). Finding the longest upsequence (resp. longest downsequence) of a permutation solves the maxi- mum independent set problem (resp. the clique problem)for the corresponding permuta- tion graph. Moreover, we discuss the problem of efficiently constructing the Young tableau for a given permutation.