  We compute the ground state correlation functions of an exactly solvable
chain of integer spins, recently introduced in [R. Movassagh and P. W. Shor,
arXiv:1408.1657], whose ground-state can be expressed in terms of a uniform
superposition of all colored Motzkin paths. Our analytical results show that
for spin s$\ge$2 there is a violation of the cluster decomposition property.
This has to be contrasted with s=1, where the cluster property holds.
Correspondingly, for s=1 one gets a light-cone profile in the propagation of
excitations after a local quench, while the cone is absent for s=2, as shown by
time dependent density-matrix-renormalization-group. Moreover, we introduce an
original solvable model of half-integer spins which we refer to as Fredkin spin
chain, whose ground-state can be expressed in terms of superposition of all
Dyck paths. For this model we exactly calculate the magnetization and
correlation functions, finding that for s=1/2, a cone-like propagation occurs
while for higher spins, s$\ge$3/2, the colors prevent any cone formation and
clustering is violated, together with square root deviation from the area law
for the entanglement entropy.
