R. P. Stanley. Two poset polytopes. Discrete Comput. Geom., 1(1):9--23, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Vol(S)Vol(S ffi ) 1=n Meyer [20] effectively showed that (S) n Vol(S) for any convex corner S. Sidorenko [25] showed that, if S is the stable set polytope of the comparability graph of a partial order P , then (S) is the number of linear extensions of P and thus, via a result of Stanley [26], the content (S) is equal to n Vol(S) We show that, if S is a 0 1 polytope, then (S) n Vol(S) if and only if S is the stable set polytope of a strongly perfect graph. The other ingredient in Meyer s proof is that (S) S ffi ) n for any convex corner S and its polar S ffi . We extend this ....

....seen, the comparability graph C(P ) of any partial order P is strongly perfect, so g (C(P ) n Vol(C(P ) In fact, as we now discuss, both sides of this identity have earlier been shown to be equal to e(P ) the number of linear extensions of P . One part of this is a 1986 result of Stanley [26]. Theorem 1.11 For any partial order P , e(P ) n Vol(Stab(C(P ) To see the direct connection between e(P ) and g (C(P ) observe that the set M(P ) of maximal elements of a partial order P is an independent set in C(P ) and that we can partition the set of linear extensions of P ....

R.Stanley, Two poset polytopes, Disc. Comp. Geom. 1 (1986) 9--23.

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R.P. Stanley, Two poset polytopes. Discrete Comput. Geom. 1 (1986), 9--23.

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R.P. Stanley, Two poset polytopes. Discrete Comput. Geom. 1 (1986), 9--23.

....to Ehrhart polynomials appears in [32, pp. 235 241] De ne the order polytope O(P ) of the nite poset P to be the set of all orderpreserving maps f : P [0; 1] fx 2 R : 0 x 1g. Thus O(P ) is a convex polytope of dimension jP j. The basic properties of order polytopes are developed in [31]. Theorem 4 Given P , C, and u as above, so u i = x 1 x i , let P i = fs 2 P C : s 6 t i 1 g (with P 1 = P C) Regard the order polytope O(P i ) as lying in R by setting coordinates indexed by elements of (P C) P i equal to 0. Then C C (P; u) x 1 O(P 1 ) x 2 O(P 2 ) ....

....(products of simplices) whose volumes are the terms in (31) Our next result concerns the combinatorial structure of the decomposition of C C (P; u) into the chambers . First we review some information from [31, x5] about the cone C(P ) of all order preserving maps f : P R 0 . The paper [31] actually deals with the order complex O(P ) rather than the cone C(P ) but this does not a ect our arguments. Recall (e.g. 32, p. 100] that an order ideal I of P is a subset of P such that if t 2 I and s t, then s 2 I. The poset (actually a distributive lattice) of all order ideals of P , ....

R. P. Stanley, Two poset polytopes, Discrete Comput. Geom. 1 (1986), 9-23.

....Ehrhart polynomials appears in [32, pp. 235 241] De ne the order polytope O(P ) of the nite poset P to be the set of all orderpreserving maps f : P [0; 1] fx 2 R : 0 x 1g. Thus O(P ) is a convex polytope in R P of dimension jP j. The basic properties of order polytopes are developed in [31]. Theorem 4 Given P , C, and u as above, so u i = x 1 x i , let P i = fs 2 P C : s 6 t i 1 g (with P 1 = P C) Regard the order polytope O(P i ) as lying in R P C by setting coordinates indexed by elements of (P C) P i equal to 0. Then C C (P; u) x 1 O(P 1 ) x 2 O(P 2 ) ....

....(products of simplices) whose volumes are the terms in (31) Our next result concerns the combinatorial structure of the decomposition of C C (P; u) into the chambers . First we review some information from [31, x5] about the cone C(P ) of all order preserving maps f : P R 0 . The paper [31] actually deals with the order complex O(P ) rather than the cone C(P ) but this does not a ect our arguments. Recall (e.g. 32, p. 100] that an order ideal I of P is a subset of P such that if t 2 I and s t, then s 2 I. The poset (actually a distributive lattice) of all order ideals of P , ....

R. P. Stanley. Two poset polytopes. Discrete Comput. Geom., 1:9-23 1986.

....polynomials appears in [32, pp. 235 241] Define the order polytope O(P ) of the finite poset P to be the set of all orderpreserving maps f : P [0; 1] fx 2 R : 0 x 1g. Thus O(P ) is a convex polytope in R P of dimension jP j. The basic properties of order polytopes are developed in [31]. Theorem 4 Given P , C, and u as above, so u i = x 1 Delta Delta Delta x i , let P i = fs 2 P Gamma C : s 6 t i Gamma1 g (with P 1 = P Gamma C) Regard the order polytope O(P i ) as lying in R P GammaC by setting coordinates indexed by elements of (P Gamma C) Gamma P i equal to ....

....(products of simplices) whose volumes are the terms in (31) Our next result concerns the combinatorial structure of the decomposition of CC (P; u) into the chambers . First we review some information from [31, x5] about the cone C(P ) of all order preserving maps f : P R0 . The paper [31] actually deals with the order complex O(P ) rather than the cone C(P ) but this does not affect our arguments. Recall (e.g. 32, p. 100] that an order ideal I of P is a subset of P such that if t 2 I and s t, then s 2 I. The poset (actually a distributive lattice) of all order ideals of P ....

R. P. Stanley. Two poset polytopes. Discrete Comput. Geom., 1:9--23 1986.

No context found.

R. P. Stanley. Two poset polytopes. Discrete Comput. Geom., 1(1):9--23, 1986.

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