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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=A0(B,N), such that p(A,n) is even for n>N. Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) considered σ(A,n)=∑d∣n,d∈Ad, and proved that for all k≥0, the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F2[z] with f(0)=1, by constructing a set A such that the generating function fA of A is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in F2[z] with P(0)=Q(0)=1, then for all k≥0 the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. [Copyright &y& Elsevier]

Abstract: In this paper we define the notion of “graded algebra with symmetries”. This notion is a generalization of the extended differential forms. We prove that for a graded algebra with symmetries T, we associate a subalgebra which generalizes the noncommutative differential forms. Using this algebra , we can define the Hochschild and cyclic homologies, cup i-products and the Steenrod squares. [Copyright &y& Elsevier]

Abstract: The fundamental example of Gerstenhaber algebra is the space of polyvector fields on , equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping algebra of a Gerstenhaber algebra . This structure gives us a definition of the Chevalley–Harrison cohomology operator for . We finally show the nontriviality of a Chevalley–Harrison cohomology group for a natural Gerstenhaber subalgebra in . [Copyright &y& Elsevier]