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 μ n , 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 .