The Fixed-point Partition Lattices

Let σ be a permutation of the set {1,2, — -, n} and let Π(N) denote the lattice of partitions of {1,2, •••,%}. There is an obvious induced action of σ on Π(N); let Π(N)σ — L denote the lattice of partitions fixed by σ. The structure of L is analyzed with particular attention paid to ^f, the meet sublattice of L consisting of 1 together with all elements of L which are meets of coatoms of L. It is shown that ^ is supersolvable, and that there exists a pregeometry on the set of atoms of ~^ whose lattice of flats G is a meet sublattice of ^ C It is shown that G is supersolvable and results of Stanley are used to show that the Birkhoff polynomials B (λ) and BG{λ) are BG{λ) = W 1)U i) U (m and