A Dynkin diagram classification theorem arising from a combinatorial problem. (English) Zbl 0639.06006
A poset P is vertex-labelable if there exists a rational-valued function \(\pi\) on P such that, for every order ideal \(I\subseteq P\), we have
\[
\sum_{x\quad \max imal\quad in\quad I}\pi (x)-| I| =\sum_{y\quad \min imal\quad in\quad P-I}\pi (y)-| P-I|.
\]
The poset of all order ideals in a vertex-labelable poset is rank symmetric, rank unimodal and strongly Sperner. The author presents a complete classification of vertex-labelable posets.
Reviewer: M.Navara
MSC:
| 06D05 | Structure and representation theory of distributive lattices |
| 06B15 | Representation theory of lattices |
| 06A06 | Partial orders, general |
Keywords:
ranked posets; strongly Sperner posets; Dynkin diagrams; order ideals; vertex-labelable poset; rank symmetric; rank unimodalReferences:
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