Generating Permutations and Combinations

We consider permutations of {1, 2, . . . , n} in which each integer is given a direction; such permutations are called directed permutations. An integer k in a directed permutation is called mobile of its arrow points to a smaller integer adjacent to it. For example, for → 3 → 2 ← 5 → 4 → 6 → 1 , the integers 3, 5, and 6 are mobile. It follows that 1 can never be mobile since there is no integer in {1, 2, . . . , n} smaller than 1. The integer n is mobile, except two cases: (i) n is the first integer and its arrow points to the left, i.e., ← n · · · ; (ii) n is the last integer and its arrow points to the right, i.e., · · · n .