Tensor algebras and displacement structure. IV. Invariant kernels
  In this paper we investigate the class of invariant positive definite kernels on the free semigroup on N generators. We provide a combinatorial description of the positivity of the kernel in terms of Dyck paths and then we find a displacement equation that encodes the invariance property of the kernel. 
T. Banks, T. Constantinescu, Nermine El-Sissi