27 research outputs found
The Reversal Ratio of a Poset
Felsner and Reuter introduced the linear extension diameter of a partially
ordered set , denoted \mbox{led}(\mathbf{P}), as the maximum
distance between two linear extensions of , 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 of 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 on at most
elements for which the reversal ratio , where
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 -element
set.Comment: 6 page
On the Stanley Depth of Squarefree Veronese Ideals
Let be a field and . In 1982, Stanley defined what is
now called the Stanley depth of an -module , denoted \sdepth(M), and
conjectured that \depth(M) \le \sdepth(M) for all finitely generated
-modules . This conjecture remains open for most cases. However, Herzog,
Vladoiu and Zheng recently proposed a method of attack in the case when with being monomial -ideals. Specifically, their method
associates with a partially ordered set. In this paper we take advantage of
this association by using combinatorial tools to analyze squarefree Veronese
ideals in . In particular, if is the squarefree Veronese ideal
generated by all squarefree monomials of degree , we show that if , then \sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d,
and if and , 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