On a conjecture of Stanley depth of squarefree Veronese ideals

Abstract

In this paper, 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 $1\le d\le n$, $\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$. Herzog, Vladoiu 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 that if $1\le d\le n\le (d+1)\lfloor \frac{1+\sqrt{5+4d}}{2}\rfloor +2d$, then $\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$. We also obtain $ \lfloor \frac{d+\sqrt{d^2+4(n+1)}}{2} \rfloor \le \sdepth(I_{n,d}) \le \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$ for $n>(d+1)\lfloor \frac{1+\sqrt{5+4d}}{2}\rfloor +2d$. As a byproduct of our construction, We give an alternative proof of Theorem $1.1$ in $[13]$ without graph theory.

11 pages; Theorem 1.2 has been changed due to a gap in the previous version

References in corpus (6)

• Stanley Conjecture in small embedding dimension
• Stanley depth of complete intersection monomial ideals
• Stanley decompositions and localization
• Stanley depth of square free Veronese ideals
• On the Stanley Depth of Squarefree Veronese Ideals
• A note on Stanley conjecture for monomial ideals

Cited by in corpus (4)

• Lexsegment ideals of Hilbert depth 1
• Gröbner bases of syzygies and Stanley depth
• Stanley depth of edge ideals
• Hilbert series and Hilbert depth of squarefree Veronese ideals