Seaweed algebras are a class of Lie algebras that are naturally characterized by a pair of compositions, which in turn are represented visually as planar graphs called meanders. These meanders provide a straightforward method for computing the index of the associated algebra. The goal of this paper is to enumerate those seaweed algebras with a fixed index and whose associated compositions have restricted part sizes. In particular, we enumerate those with composition part sizes from so-called acyclic sets. We also establish a bijection between sets of indecomposable seaweed algebras with meanders with certain restricted part sizes and sets of permutations with restricted displacements. In certain cases, the index of the algebra can be determined by a simple statistic on the permutation.