Give a counterexample showing that the number of compositions of n into k parts does not always match the number of partitions of n into k parts. For which n and k do they differ?
For \(n=4\) and \(k=2\) the compositions are 1+3, 2+2 and 3+1 (3 cases), but the partitions are only 1+3 and 2+2 (2 cases), since 1+3 and 3+1 count once. The one with \(n=3\), \(k=3\) gives 3 compositions when only 1+1+1 exists, and the one with \(n=4\), \(k=1\) shows no difference at all.