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We compute dimensions of the components for the operad of two compatiblebrackets and for the bihamiltonian operad. We also obtain character formulasfor the representations of the symmetric groups and the $SL_2$ group in thesespaces.Comment: 24 pages, accepted by Functional Analysis and its Applications, a few  typos correcte

Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad