划分格中的Mbius函数和秩生成函数

Mobius functions and Rank-generating functions in Finite Fields

设S={1,2,…,n},P(n)是由S的所有划分组成的集合.对于π,σ∈P(n),如果π中的每个块包含在σ的一个块里,就定义π≤σ,那么P(n)作成一个格.如果M(n,k)是由S的所有k部划分组成的集合,而L(n,k)是由M(n,k)生成的格.在P(n)和L(n,k)中,给出M(o|¨)bius函数,并且确定了特征多项式和秩生成函数的表示式.

Suppose that S = {1,2,…,n} and the set that is of consist of all partition of S,denote by P（n）.For π,σ ∈P（n）,together with the condition of every block belong to π is also included in any block of σ,then we define π≤σ and P（n） is known as a lattice.If the set M{n,k） that is of consist of all k-partition of S,and the lattice generated by M（n,k） by L（n,k）.In the P（n） and L（n,k） the Mobius functions and is given,and the expressions of the characteristic polynomials and the rank-generating functions are obtained are also determined.