Weighted Compositions

Abstract

By a composition of a positive integer n is meant a representation of n as a sum of one or more positive integers where the order of the summands is taken into account. Thus for example 4 has the eight compositions 4 = 3 + 1 = 1 +3 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1. Now n can be written in the form 1 + 1 + … + 1 with n - 1 plus signs. Deletion of any subset of these plus signs breaks n into parts which form a composition of n. Conversely, any composition of n corresponds to a subset of plus signs, so that the number of compositions of n is the number of subsets of a set with n - 1 elements, namely 2n - 1. In this note we obtain a number of generalizations of this rather obvious remark by making use of the notion of a weighted composition and the method of generating series.