Interval partitions and Stanley depth

In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1,2,…,n} into intervals. Specifically, given a positive integer n, they asked whether there exists a partition P(n) of the non-empty subsets of {1,2,…,n} into intervals, so that |B|⩾n/2 for each interval [A,B] in P(n). We answer this question in the affirmative by first embedding it in a stronger result. We then provide two alternative proofs of this second result. The two proofs use entirely different methods and yield non-isomorphic partitions. As a consequence, we establish that the Stanley depth of the ideal (x1,…,xn)⊆K[x1,…,xn] (K a field) is ⌈n/2⌉.