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Two poset polytopes. (English) Zbl 0595.52008
With a partially ordered set \(P\) with \(n\) elements, the author associates two \(n\)-dimensional convex polytopes, the order polytope \(\mathcal O(P)\) and the chain polytope \(\mathcal C(P)\). He determines the face lattice of \(\mathcal O(P)\), the vertices of \(\mathcal C(P)\), and describes a piecewise-linear bijection from \(\mathcal O(P)\) onto \(\mathcal C(P)\) which allows to transfer properties of \(\mathcal O(P)\) over to \(\mathcal C(P)\). It is shown that the Ehrhart polynomials of \(\mathcal O(P)\) and \(\mathcal C(P)\) satisfy \(i(\mathcal O(P),m)=i(\mathcal C(P),m)=\Omega (P,m+1)\), where \(\Omega(P,m)\), the order polynomial, is the number of order-preserving maps \(P\to \{1,\ldots,m\}\). In particular, \(n!\,\mathrm{vol}\,\mathcal O(P)= n!\,\mathrm{vol}\,\mathcal C(P)\) is the number of linear extensions of \(P\). Similarly as in a former paper [J. Comb. Theory Ser. A 31, 56–65 (1981; Zbl 0484.05012)], the author uses the Aleksandrov-Fenchel inequalities for mixed volumes in connection with \(\mathcal C(P)\) to obtain new log-concave sequences involving linear extensions of \(P\).

MSC:
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
06A06 Partial orders, general
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References:
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