Symplectic F_2-vector spaces and Dyck words

Let $V$ be an $F_2$-vector space with a given nondegenerate symplectic form and a circular basis. We define a partition of $V$ into subsets each of which has cardinal equal to a product of Catalan numbers. This can be viewed as a partition of the set of unipotent representations of a finite symplectic group corresponding to a certain two-sided cell.