A composition of n into s parts is a partition in which the order of the parts is taken into account. For example, the compositions of 4 into two parts are (3, 1), (1, 3), (2, 2). In the above example there is only one composition of 4 into two parts with no part exceeding 2, namely (2, 2). Let c(n, s, r ) be the number of compositions of n into s parts with no part exceeding r, and let c(s, r ) be the maximum of c(n, s, r ) over n for fixed s and r. The following results are established for r ~> 2: