XVII. Memoir on the theory of the compositions of numbers

Abstract

Abstract 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 xn in the expansion of (x + x2 + x3 + . . .)p; this expression may be written (x/1-x)p, and the coefficient of xn is seen to be (n-1 p-1).