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 ( x 1 , … , x n ) ⊆ K [ x 1 , … , x n ] (K a field) is ⌈ n / 2 ⌉ .