On the Stanley depth of squarefree Veronese ideals

Keller, Mitchel T. ; Shen, Yi-Huang ; Streib, Noah ; Young, Stephen J.

Journal of Algebraic Combinatorics, Tome 33 (2011) no. 2, pp. 313-324.

Résumé

Summary: Let $K$ be a field and $S= K[ x _{1},\cdots , x _{ n }]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted sdepth $( M)$, and conjectured that depth $( M)\leq $sdepth $( M)$ for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M= I/ J$ with $J\subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I _{ n, d }$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1\leq d\leq $n$5 d+4$, then sdepth $( I _{ n, d })=\lfloor $( n - d)/$( d+1)\rfloor + d$, and if $d\geq 1$ and $n\geq 5 d+4$, then $d+3\leq $sdepth $( I _{ n, d })\leq \lfloor $( n - d)/$( d+1)\rfloor + d$.

Détail

Keywords: keywords Stanley depth, squarefree monomial ideal, interval partition, squarefree Veronese ideal