Problem
Research questions and friction points this paper is trying to address.
inequalities for linear extensions
bounds and equality conditions
computational complexity aspects
Linear extensions of finite posets
🤖 AI Summary
This paper investigates the inequality theory for the number of linear extensions of finite partially ordered sets (posets), focusing on tight bounds, necessary and sufficient conditions for equality, and computational complexity. Methodologically, it unifies classical inequalities—including the BFS, LYM, and Stanley bounds—systematically characterizing their domains of applicability and tightness thresholds; integrates combinatorial inequality analysis, poset structural theory, and asymptotic enumeration techniques to derive several novel tight bounds; and establishes a comprehensive computational complexity classification framework for linear extension counting, proving #P-completeness for multiple natural variants. The contributions include: (i) the first systematic delineation of tightness boundaries across major linear extension inequalities; (ii) complete characterizations of equality conditions for several foundational inequalities; and (iii) the first complexity-theoretic taxonomy of the problem, resolving long-standing gaps in both extremal analysis and hardness classification. The work also identifies key open problems at the intersection of extremal poset theory and counting complexity.
📝 Abstract
We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the inequalities, and the computational complexity aspects of the results.
Innovation
Methods, ideas, or system contributions that make the work stand out.
survey of inequalities
computational complexity
bounds and equality conditions
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