Abstract :
Let Πn, k be the partially ordered set whose elements are all nonempty intersections of the affine hyperplanes Hi,j,r={x∈Rn:xi=xj+r} for integers i, j, k, r such that 1 ⩽ i, j ⩽ n and vbr ⩽ k, ordered by reverse inclusion. First we show that for a fixed k, the exponential generating function Mk(x) of the number of maximal elements in this poset is Mk(x)=ex−1(1+k)−kex, and then from this, it follows immediately, using species, that the number of elements in this poset which have a given dimension d is the coefficient of tdxn/n! in Nk(x, t) = etMk(x).
After we do this, we use the fact that Mk+1(x) can be expressed in terms of Mk(x) for each k to show that this implies that there is a bijection between the set of maximal elements of Πn, k+1 and a certain other set.
