The lattice of finite subspace partitions

Let VV denote V(n,q)V(n,q), the vector space of dimension nn over GF(q)(q). A subspace partition   of VV is a collection ΠΠ of subspaces of VV such that every nonzero vector in VV is contained in exactly one subspace belonging to ΠΠ. The set P(V)P(V) of all subspace partitions of VV 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)P(V) is congruent to the number of all set partitions of {1,…,n}{1,…,n} modulo q−1q−1. Moreover, we show that the Möbius number μn,q(0,1) of P(V)P(V) is congruent to (−1)n−1(n−1)!(−1)n−1(n−1)! (the Möbius number of set partitions of {1,…,n}{1,…,n}) modulo q−1q−1.