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Let P be a poset in which each point is incomparable to at most ∆ others. Tanen-baum, 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… (More)

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |h L (x) − h L (y)| ≤ k, where h L (x) is the height of x in L. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing… (More)

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P) ≤ d. We answer this question in the affirmative. We also show that if P is a poset containing the standard example S 5 as a subposet, then the cover graph of P has treewidth at least… (More)

th birthday. Among his many remarkable contributions to combinatorial mathematics and theoretical computer science is a jewel for online problems for partially ordered sets: the fact that h(h + 1)/2 antichains are required for an online antichain partition of a poset of height h. The linear discrepancy of a poset P is the least k for which there is a linear… (More)

a r t i c l e i n f o a b s t r a c t In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1, 2,. .. ,n} into intervals. Specifically,… (More)

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