[Paper Review] On a conjecture of Stanley depth of squarefree Veronese ideals

Maorong Ge, Jiayuan Lin+1 more|arXiv (Cornell University)|Nov 29, 2009

Commutative Algebra and Its Applications13 references4 citations

TL;DR

This paper partially confirms a conjecture on the Stanley depth of squarefree Veronese ideals $I_{n,d}$, proving that $\operatorname{sdepth}(I_{n,d}) = \left\lfloor \binom{n}{d+1}/\binom{n}{d} \right\rfloor + d$ when $n \leq (d+1)\left\lfloor \frac{1+\sqrt{5+4d}}{2} \right\rfloor + 2d$, and establishes bounds for larger $n$. The authors modify combinatorial interval partitioning techniques using higher circular representations to construct Stanley decompositions, offering a graph-theory-free proof of a prior result.

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

Motivation & Objective

• To verify a conjecture on the Stanley depth of squarefree Veronese ideals $I_{n,d}$ for certain ranges of $n$ and $d$.
• To develop a combinatorial method based on interval partitions of posets associated with monomial ideals.
• To provide an alternative proof of Theorem 1.1 in [13] without relying on graph theory.
• To establish tight bounds for the Stanley depth when $n$ exceeds a critical threshold dependent on $d$.

Proposed method

• Adapt the poset-based method of Herzog, Vladoiu, and Zheng to relate Stanley depth to interval partitions of the support poset of $I_{n,d}$.
• Introduce a higher circular representation to construct interval partitions that yield improved Stanley depth estimates.
• Use double induction on $d$ and $n$ to prove lower bounds on $\operatorname{sdepth}(I_{n,d})$ for $n \geq (d+1)k + d$, leveraging known base cases.
• Construct interval families $\mathscr{I}_{n,d,k+1}$ and $\mathscr{I}_{n,d+l,k-l+1}$ to cover subsets of various sizes in the poset.
• Ensure disjointness of intervals via Propositions 3.3 and 3.5, guaranteeing valid Stanley decompositions.
• Combine the constructed partitions with trivial intervals to cover all uncovered elements, yielding a complete interval partition of $P_{I_{n,d}}$.

Experimental results

Research questions

• RQ1 Does the conjectured formula $\operatorname{sdepth}(I_{n,d}) = \left\lfloor \binom{n}{d+1}/\binom{n}{d} \right\rfloor + d$ hold for all $1 \leq d \leq n$?
• RQ2 Can the Stanley depth of $I_{n,d}$ be bounded below and above using combinatorial interval partitioning techniques?
• RQ3 Is it possible to prove Theorem 1.1 in [13] without using graph-theoretic tools?
• RQ4 What is the behavior of $\operatorname{sdepth}(I_{n,d})$ when $n > (d+1)\left\lfloor \frac{1+\sqrt{5+4d}}{2} \right\rfloor + 2d$?
• RQ5 Can the construction of interval partitions be generalized to yield tight bounds beyond the known conjectured formula?

Key findings

• For $n \leq (d+1)\left\lfloor \frac{1+\sqrt{5+4d}}{2} \right\rfloor + 2d$, the Stanley depth of $I_{n,d}$ equals $\left\lfloor \binom{n}{d+1}/\binom{n}{d} \right\rfloor + d$, confirming the conjecture in this range.
• When $n > (d+1)\left\lfloor \frac{1+\sqrt{5+4d}}{2} \right\rfloor + 2d$, the Stanley depth satisfies $\left\lfloor \frac{d + \sqrt{d^2 + 4(n+1)}}{2} \right\rfloor \leq \operatorname{sdepth}(I_{n,d}) \leq \left\lfloor \binom{n}{d+1}/\binom{n}{d} \right\rfloor + d$, establishing tight bounds.
• The authors provide a graph-theory-free proof of Theorem 1.1 in [13], relying on interval partitioning and induction instead.
• The construction confirms $\operatorname{sdepth}(I_{(d+1)k+d,d}) = d + k$ for $k = 0,1,2,3$, extending the validity of the lower bound to $k=3$.
• The method yields a complete interval partition of $P_{I_{n,d}}$ by combining non-overlapping intervals from multiple families and adding trivial intervals for uncovered elements.
• The lower bound $\left\lfloor \frac{d + \sqrt{d^2 + 4(n+1)}}{2} \right\rfloor$ is achieved via a recursive selection process of intervals with increasing size, ensuring minimal right endpoint size is maximized.