2009/11/29 by Maorong Ge, Ge, Maorong, Jiayuan Lin +3
Computer Science · Mathematics · #05E40 #06A07 #13C13 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.CO #msc:05E40 #msc:06A07 #msc:13C13
paper · pdf · doi:10.48550/arxiv.0911.5458
11 pages; Theorem 1.2 has been changed due to a gap in the previous version
openalex publication_date 2009/11/29 · arxiv created 2010/01/27 · arxiv updated 2010/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we partially confirm a conjecture, proposed by Cimpoeaş, Keller, Shen, Streib and Young, on the Stanley depth of squarefree Veronese ideals In,d. This conjecture suggests that, for positive integers 1 ≤ d ≤ n, \sdepth (In,d)= \lfloor \binomnd+1/\binomnd \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 ≤ d ≤ n ≤ (d+1) \lfloor (1+√(5+4d))/(2)\rfloor+2d, then \sdepth (In,d)= \lfloor \binomnd+1/\binomnd \rfloor+d. We also obtain \lfloor (d+√(d2+4(n+1)))/(2) \rfloor ≤ \sdepth(In,d) ≤ \lfloor \binomnd+1/\binomnd \rfloor+d for n > (d+1) \lfloor (1+√(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.