An Analogue of Young’s Lattice for Compositions

Let Cn = {compositions of n}, C = ∪Cn. We define a partial order making C into a ranked poset having 1 as its bottom element and Cn as its (n− 1)-st rank level. Let α = a1 + · · · + ak ∈ Cn. The interval [1, α] is shown to have the following properties: • The number of maximal chains in [1, α] equals the number of permutations of [n] with descent set {a1, a1 + a2, . . .}. • The interval [1, α] is CL-shellable. • The Möbius function satisfies μ(1, α) = { (−1)n−1 if α = x22 . . . 22y, x, y ∈ {1, 2}, 0 otherwise. Furthermore, there is a Pieri-type rule