Chain Polytope

The chain polytope of a finite partially ordered set is the convex polytope

$C(P)={x in R^P:x_p>=0, sum_(p in C)x_p<=1 for every chain C subset= P}.$

It suffices to impose the inequalities for maximal chains. The vertices are the coordinate vectors of the indicator functions of antichains, including the empty antichain (Stanley 1986).

If is an -element chain, then is the simplex defined by and . If is an -element antichain, it is the unit hypercube.

Stanley's (1986) transfer map from the order polytope to the chain polytope is

$y_p=x_p-max({x_q:q<_Pp} union {0}).$

This is a piecewise linear bijection. The two polytopes have the same Ehrhart polynomial, volume, and number of vertices.

Their 2-dimensional faces are triangles or quadrilaterals. The quadrilateral counts agree, while the triangular count for the chain polytope is at least that for the order polytope (Freij-Hollanti et al. 2026).

See also

Antichain, Chain, Ehrhart Polynomial, Order Polytope

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References

Freij-Hollanti, R.; Lundström, T.; and Mori, A. "Two-Dimensional Faces of Order and Chain Polytopes." Electron. J. Combin. 33, P3.67, 2026. https://doi.org/10.37236/14909.Stanley, R. P. "Two Poset Polytopes." Discrete Comput. Geom. 1, 9-23, 1986. https://doi.org/10.1007/BF02187680.

Cite this as:

Weisstein, Eric W. "Chain Polytope." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChainPolytope.html

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