On the counting function of the sets of parts AA such that the partition function p(A,n)p(A,n) takes even values for nn large enough

If AA is a set of positive integers, we denote by p(A,n)p(A,n) the number of partitions of nn with parts in AA. First, we recall the following simple property: let f(z)=1+∑n=1∞εnzn be any power series with εn=0εn=0 or 11; then there is one and only one set of positive integers A(f)A(f) such that p(A(f),n)≡εn(mod2) for all n≥1n≥1. Some properties of A(f)A(f) have already been given when ff 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}A(P,x)=Card{a∈A(P);a⩽x} when PP is a polynomial with coefficients 00 or 11, and P(0)=1P(0)=1.