The dominance order for permutations

We define an order relation over S_n considering the Robinson-Schensted bijection and the dominance order over Young tableaux. This order relation makes S_n(k k - 1...3 2 1) -the set of permutations of length n that avoid the pattern k k - 1...3 2 1, k ≤ n- a principal filter in Sn. We study in detail these order relations on Sn(321) and Sn(4321), finding order-isomorphisms between these sets and sets of lattice paths.