Subspace partitions of Fqn containing direct sums

Let q be a prime power and n be a positive integer. A subspace partition of V=Fqn, the vector space of dimension n over Fq, is a collection Π of subspaces of V such that each nonzero vector of V is contained in exactly one subspace in Π; the multiset of dimensions of subspaces in Π is then called a Gaussian partition of V. We say that Πcontains a direct sum if there exist subspaces W1,…,Wk∈Π such that W1⊕⋯⊕Wk=V. In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers a1 and a2 with n>a1>a2≥1, our main theorem shows that if Π is a subspace partition of Fqn with mi subspaces of dimension ai for i=1,2, then Π contains a direct sum when a1x1+a2x2=n has a solution (x1,x2) for some integers x1,x2≥0 and m2 belongs to the union I of two natural intervals. The lower bound of I captures all subspace partitions with dimensions in {a1,a2} that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when m2∉I or when the condition on the existence of a nonnegative integral solution (x1,x2) is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when q→1) as well as subspace and set partitions.