Vector spaces, linear dependence, rank, lineability, Combinatorial aspects of partitions of integers, characteristic subspaces, Gaussian binomial coefficients, generating polynomials, combinatorics, hyperinvariant subspaces, MATEMATICA APLICADA, Factorials, binomial coefficients, combinatorial functions, We obtain the cardinality of the lattice of characteristic sub-spaces of a nilpotent Jordan matrix when the underlying field is GF(2), the only field where the lattices of characteristic and hyperinvariant subspaces can be different. If the characteristic polynomial of the matrix splits in the field, the general case can be reduced to the nilpotent Jordan case. Results are complex and highly combinatorial, and include the design of an algorithm.