# Fonctions de partitions à parité périodique

- Authors
- Journal
- European Journal of Combinatorics 0195-6698
- Publisher
- Elsevier
- Publication Date
- Volume
- 24
- Issue
- 8
- Identifiers
- DOI: 10.1016/s0195-6698(03)00115-x

## Abstract

Abstract Let N be the set of positive integers and A a subset of N . For n∈ N , let p( A,n) denote the number of partitions of n with parts in A . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any N∈ N and B⊂{1,2,…,N} , there is a unique set A= A 0(B,N) , such that p( A,n) is even for n> N. Soon after, Ben Saı̈d and Nicolas (Acta Arith. 106 (2003) 183) considered σ( A,n)=∑ d∣n,d∈ A d , and proved that for all k≥0, the sequence (σ( A,2 kn) mod 2 k+1) n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F 2[z] with f(0)=1, by constructing a set A such that the generating function f A of A is congruent to f modulo 2, and by showing that if f= P/ Q, where P and Q are in F 2[z] with P(0)= Q(0)=1, then for all k≥0 the sequence (σ( A,2 kn) mod 2 k+1) n≥1 is periodic on n.

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