Counting the Restricted Gaussian Partitions of a Finite Vector Space

A subspace partition Pi of a finite vector space V V(n, q) of dimension n over GF(q) is a collection of subspaces of V such that their union is V, and the intersection of any two subspaces in Pi is the zero vector. The multiset T-H of dimensions of subspaces in Pi is called the type of Pi, or a Gaussian partition of V. Previously, we showed that subspace partitions of V and their types are natural, combinatorial q-analogues of the set partitions of (1,..., n} and integer partitions of n respectively. In this paper, we connect all four types of partitions through the concept of "basic" set, subspace, and Gaussian partitions, corresponding, to the integer partitions of n. In particular, we combine Beutelspacher's classic construction of subspace partitions with some additional conditions to derive a special subset g of Gaussian partitions of V. We then show that the cardinality of g is a rational polynomial R(q) in q, with R(1) = p(n), where p is the integer partition function.