The Action of the Symmetric Group on a Generalized Partition Semilattice

Given an integer n 2, and a non-negative integer k, consider all affine hyperplanes in Rn of the form xi = xj +r for i, j [n] and a non-negative integer r k. Let n,k be the poset whose elements are all nonempty intersections of these affine hyperplanes, ordered by reverse inclusion. It is noted that n,0 is isomorphic to the well-known partition lattice n, and in this paper, we extend some of the results of n by Hanlon and Stanley to n,k. Just as there is an action of the symmetric group Sn on n, there is also an action on n,k which permutes the coordinates of each element. We consider the subposet n,k of elements that are fixed by some Sn, and find its M

• Type: Journal
• Year: 2000
• Where: COMBINATORICS
• Authors: Robert Gill
• Keywords: COMBINATORICS 2000, Non-negative Integer, Well-known Partition Lattice, Xj +r