1. Compositions are merely partitions in which the order of occurrence of the parts is essential; thus, while the partitions of the number 3 are (3), (21), (12), (111). The enumerations of the compositions of a number n into p parts, zeros excluded, is given by the coefficient of x n in the expansion of ( x + x 2 + x 3 + . . .) p ; this expression may be written ( x /1- x ) p , and the coefficient of x n is seen to be ( n -1 p -1).