On the counting function of the sets of parts [formula omitted] such that the partition function [formula omitted] takes even values for [formula omitted] large enough

Abstract

If A is a set of positive integers, we denote by p ( A , n ) the number of partitions of n with parts in A . First, we recall the following simple property: let f ( z ) = 1 + ∑ n = 1 ∞ ε n z n be any power series with ε n = 0 or 1 ; then there is one and only one set of positive integers A ( f ) such that p ( A ( f ) , n ) ≡ ε n ( mod 2 ) for all n ≥ 1 . Some properties of A ( f ) have already been given when f is a polynomial or a rational fraction. Here, we give some estimations for the counting function A ( P , x ) = Card { a ∈ A ( P ) ; a ⩽ x } when P is a polynomial with coefficients 0 or 1 , and P ( 0 ) = 1 .