On a Conjecture of Stanley Depth of Squarefree Veronese Ideals

Abstract

In this article, we partially confirm a conjecture, proposed by Cimpoeaş, Keller, Shen, Streib, and Young, on the Stanley depth of squarefree Veronese ideals I n, d . This conjecture suggests that, for positive integers 1 ≤ d ≤ n, . Herzog, Vlădoiu, and Zheng established a connection between the Stanley depths of quotients of monomial ideals and interval partitions of certain associated posets. Based on this connection, Keller, Shen, Streib, and Young recently developed a useful combinatorial tool to analyze the interval partitions of the posets associated with the squarefree Veronese ideals. We modify their ideas and prove the above conjecture for . We also obtain a lower bound of sdepth(I n, d ) for any 1 ≤ d ≤ n. Our results greatly improve Theorem 1.1 in [13], and moreover, our construction leads to a direct proof of this theorem without using graph theory.