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Abstract  

For P

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

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Summary  

For PF 2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of  N (cf. [9]) such that Σn 0 p(A,n)z n  ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d |n, d A d, namely, the periodicity of the sequences (σ(A,2k n) mod 2k +1)n 1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2k n) mod 2k +1 will be established, from which it will be shown that the weight σ(A1,2k z i) mod 2k +1    on the orbit \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $z_i$ \end{document} is moved on some other orbit z j when A1 is replaced by A2 with A1= A(P 1) and A2= A(P 2) P 1 and P 2 being irreducible in F 2[z]  of the same odd order.

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