A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; (3) the reduced automaton is used to compute an expression (in terms of stream constants and operators) that represents the stream of all counts.

Let M(A, θ) be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet B ⊂ A such that the right factor of a bisection M(A, θ) = M(B, θB).T be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of M(A, θ) and associated bases of LK(A, θ). Résumé Soit M(A, θ) un monoïde partiellement commutatif libre. Nous donnons une condition nécessaire et suffisante sur un sous alphabet B ⊂ A pour que le facteur droit d’une bisection de la forme M(A, θ) = M(B, θB).T soit partiellement commutatif libre. Ceci nous permet d’étendre strictement et de façon optimale la théorie (classique) de l’élimination avec commutations partielles et de construire de nouvelles factorisations de M(A, θ) ainsi que les bases de LK(A, θ) associées.