A Kruskal–Katona type theorem for integer partitions

Let N be the set of positive integers, and let P ( n ) = ⋃ 1 ≤ l ≤ n { ( x 1 , … , x l ) ∈ N l : x 1 + ⋯ + x l = n } be the set of (ordered) partitions of n . We show that there exist a rank function and orderings ≤ c and ≺ such that the ranked poset ( P ( n ) , ≤ c , ≺ ) is Macaulay.