##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Stephen Young

#### UCSD

## Stanley Depth and Interval Partitions

##### Abstract:

In 1982 Stanley developed a geometric/combinatorial invariant for a finitely generated $\mathbb{Z}^n$-graded module $M$ over a polynomial ring in $n$ variables and conjectured that this quantity was an upper bound on the depth of $M$. Since then relatively little progress has been made in resolving Stanley's conjecture, in part because of the difficulty of calculating the Stanley depth. Recently, Herzog, Vladoiu, and Zheng provided an algorithm for calculating the Stanley depth of the quotient of monomial ideas over a polynomial ring. We use their result to resolve the Stanley depth of the maximal square-free monomial ideal and end up with a previously unknown property of the boolean lattice. We extend these ideas to partially resolve the case of square-free Veronese ideals. \\ This is joint work with Csaba Bir$\mathrm{\acute{o}}$, Dave Howard, Mitch Keller, Noah Streib, Yi-Huang Shen, and Tom Trotter.

### March 2, 2010

### 3:00 PM

### AP&M 7321

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