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

Zbl 0484.05012
Full Text:

References:

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