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Order-chain polytopes. (English) Zbl 1419.52010

Summary: Given two families \(X\) and \(Y\) of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate a new class of polytopes is to take the intersection \(\mathcal P = \mathcal P_1 \cap \mathcal P_2\), where \(\mathcal P_1 \in X,\ \mathcal P_2 \in Y\). Two basic questions then arise: 1) when \(\mathcal P\) is integral and 2) whether \(\mathcal P\) inherits the “old type” from \(\mathcal P_1,\ \mathcal P_2\) or has a “new type”, that is, whether \(\mathcal P\) is unimodularly equivalent to a polytope in \(X \cup Y\) or not. In this paper, we focus on the families of order polytopes and chain polytopes. Following the above framework, we create a new class of polytopes which are named order-chain polytopes. When studying their volumes, we discover a natural relation with Ehrenborg and Mahajan’s results on maximizing descent statistics.

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Software:

polymake
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Full Text: DOI arXiv

References:

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