Sets with even partition functions and 2-adic integers

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For P ∊ \( \mathbb{F}_2 \)[z] with P(0) = 1 and deg(P) ≥ 1, let \( \mathcal{A} \) = \( \mathcal{A} \)(P) (cf. [4], [5], [13]) be the unique subset of ℕ such that Σn≥0p(\( \mathcal{A} \), n)zn ≡ P(z) (mod 2), where p(\( \mathcal{A} \), n) is the number of partitions of n with parts in \( \mathcal{A} \). Let p be an odd prime and P ∊ \( \mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + zp in \( \mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \( \mathcal{A} \) = \( \mathcal{A} \)(P) of the form 2km are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Said and J.-L. Nicolas [6] to all primes p.