Genuinely entangled symmetric states with no N-partite correlations

We investigate genuinely entangled N-qubit states with no N-partite correlations in the case of symmetric states. Using a tensor representation for mixed symmetric states, we obtain a simple characterization of the absence of N-partite correlations. We show that symmetric states with no N-partite correlations cannot exist for an even number of qubits. We fully identify the set of genuinely entangled symmetric states with no N-partite correlations in the case of three qubits, and in the case of rank-2 states. We present a general procedure to construct families for an arbitrary odd number of qubits.