Let and be, respectively, the number of subsets and -subsets of such that no two subset elements differ by an element of the set , the largest element of which is . We prove a bijection between such -subsets when with and permutations of with excedances satisfying for all . We also identify a bijection between another class of restricted permutation and the cases and derive the generating function for when . We give some classes of for which is also the number of compositions of into a given set of allowed parts. We also prove a bijection between -subsets for a class of and the set representations of size of equivalence classes for the occurrence of a given length-() subword within bit strings. We then formulate a straightforward procedure for obtaining the generating function for the number of such equivalence classes.