K-sorted Permutations with Weakly Restricted Displacements

A permutation ) ( 2 1 n π π π π  = of {1, 2,..., n} is called k-sorted if and only if , k i i ≤ −π for all . 1 n i ≤ ≤ We propose an algorithm for generating the set of all k-sorted permutations of {1, 2,..., n} in lexicographic order. An inversion occurs between a pair of ( i π , j π ) if i < j but i π > j π . Let I(n, k) denote the maximum number of inversions in k-sorted permutations. For k-sorted permutations with weakly restricted displacements, i.e.,   , 1 2 / − ≤ ≤ n k n we propose a concise formula of I(n, k) by using the generating functions approach.