The lattice of finite subspace partitions

Let V denote V (n, q), the vector space of dimension n over GF(q). A subspace partition of V is a collection Π of subspaces of V such that every nonzero vector in V is contained in exactly one subspace belonging to Π. The set P(V ) of all subspace partitions of V is a lattice with minimum and maximum elements 0 and 1 respectively. In this paper, we show that the number of elements in P(V ) is congruent to the number of all set partitions of {1, . . . , n} modulo q − 1. Moreover, we show that the Möbius number μn,q(0,1) of P(V ) is congruent to (−1)n−1(n − 1)! (the Möbius number of set partitions of {1, . . . , n}) modulo q − 1.