In this section we will derive basic results on compositions, and also look at some variations of compositions, by restricting either the placement of the parts within the composition or the set from which the parts are taken. Examples of restriction in terms of placement of the parts are Carlitz compositions and palindromic compositions. In the area of restricted sets, we will present the results of Alladi and Hoggatt [7] and Grimaldi [76], who studied compositions in {1, 2} and compositions with odd parts. We will then derive results for a general set A which includes their results as well as results on the sets A = {1, k} [50], A = N\{1} [77], A = N\{2} [52], and more generally, A = N\{k} [51] as special cases.