Gaussian partitions

Let V=V(n,q) denote the finite vector space of dimension n over the finite field with q elements. A subspace partition of V is a collection Π of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. In a recent paper, we proved some strong connections between the lattice of the subspace partitions of V and the lattice of the set partitions of n={1,…,n}. We now define a Gaussian partition of [n] _q =(q ⁿ −1)/(q−1) to be a nonincreasing sequence of positive integers formed by ordering all elements of some multiset {dim(W):W∈Π}, where Π is a subspace partition of V. The Gaussian partition function gp(n,q) is then the number of all Gaussian partitions of [n] _q , and is naturally analogous to the classical partition function p(n). In this paper, we initiate the study of gp(n,q) by exhibiting all Gaussian partitions for small n. In particular, we determine gp(n,q) as a polynomial in q for n≤5, and find a lower bound for gp(6,q).