Sur les ensembles représentés par les partitions dʹun entier n1 Original Research Article

Let n = n1 + n2 + … + nj a partition Π of n. One will say that this partition represents the integer a if there exists a subsum nil + ni2 + … + nil equal to a. The set E(Π) is defined as the set of all integers a represented by Π. Let A be a subset of the set of positive integers. We denote by p(A,n) the number of partitions of n with parts in A, and by p̌ ((A,n) the number of distinct sets represented by these partitions. Various estimates for p̌ (A,n) are given. Two cases are more specially studied, when A is the set {1, 2, 4, 8, 16, …} of powers of 2, and when A is the set of all positive integers. Two partitions of n are said to be equivalent if they represent the same integers. We give some estimations for the minimal number of parts of a partition equivalent to a given partition.