The Reversal Ratio of a Poset

Felsner and Reuter introduced the linear extension diameter of a partially ordered set $\mathbf{P}$, denoted \mbox{led}(\mathbf{P}), as the maximum distance between two linear extensions of $\mathbf{P}$, where distance is defined to be the number of incomparable pairs appearing in opposite orders (reversed) in the linear extensions. In this paper, we introduce the reversal ratio $RR(\mathbf{P})$ of $\mathbf{P}$ as the ratio of the linear extension diameter to the number of (unordered) incomparable pairs. We use probabilistic techniques to provide a family of posets $\mathbf{P}_k$ on at most $k\log k$ elements for which the reversal ratio $RR(\mathbf{P}_k)\leq C/\log k$, where $C$ is a constant. We also examine the questions of bounding the reversal ratio in terms of order dimension and width.Comment: 10 pages, 2 figures; Accepted for publication in ORDE

Asymptotic Enumeration of Labelled Interval Orders

Building on work by Zagier, Bousquet-M\'elou et al., and Khamis, we give an asymptotic formula for the number of labelled interval orders on an $n$-element set.Comment: 6 page

On the Stanley Depth of Squarefree Veronese Ideals

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted \sdepth(M), and conjectured that \depth(M) \le \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 $1\le d\le n < 5d+4$, then \sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d, and if $d\geq 1$ and $n\ge 5d+4$, then d+3\le \sdepth(I_{n,d}) \le \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d.Comment: 10 page

Degree Bounds for Linear Discrepancy of Interval Orders and Disconnected Posets

Let P be a poset in which each point is incomparable to at most ∆ others. Tanenbaum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by ⌊(3 ∆ − 1)/2⌋. In this paper, we answer this question in the affirmative for two classes of posets by proving upper bounds in terms of ∆. We first prove a Brooks-type bound on the linear discrepancy of interval orders, and because of the equivalence of the linear discrepancy of a poset and the bandwidth of its co-comparability graph, we obtain the same bound for the bandwidth of interval graphs. Specifically, the linear discrepancy of an interval order is at most ∆, with equality if and only if it contains an antichain of size ∆ + 1. Furthermore, the stronger bound is tight even for interval orders of width 2. The second class of posets we consider are the disconnected posets, which we show have linear discrepancy at most ⌊(3 ∆ − 1)/2⌋. To facilitate the proofs of these results, we also prove lemmas on the role of critical pairs in linear discrepancy as well as a theorem establishing every poset contains a point whose removal decreases the linear discrepancy by at most 1