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Download Download S-hypersimplices, pulling triangulations, and monotone paths Sebastian Manecke Raman Sanyal Jeonghoon So Institut f¨urMathematik Goethe-Universit¨atFrankfurt Germany [email protected] [email protected] [email protected] Submitted: Jan 17, 2019; Accepted: May 28, 2020; Published: Jul 24, 2020 c The authors. Released under the CC BY-ND license (International 4.0). Abstract An S-hypersimplex for S ⊆ f0; 1; : : : ; dg is the convex hull of all 0=1-vectors of length d with coordinate sum in S. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of S-hypersimplices. Moreover, we show that monotone path polytopes of S-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order. Mathematics Subject Classifications: 52B20, 52B12 1 Introduction d The cube d = [0; 1] together with the simplex ∆d = conv(0; e1;:::; ed) and the cross-polytope ♦d = conv(±e1;:::; ±ed) constitute the Big Three, three infinite families of convex polytopes whose geometric and combinatorial features make them ubiquitous throughout mathematics. A close cousin to the cube is the (even) halfcube d Hd := conv p 2 f0; 1g : p1 + ··· + pd even : d The halfcubes H1 and H2 are a point and a segment, respectively, but for d > 3, Hd ⊂ R is a full-dimensional polytope. The 5-dimensional halfcube was already described by Thomas Gosset [11] in his classification of semi-regular polytopes. In contemporary mathematics, halfcubes appear under the name of demi(hyper)cubes [7] or parity polytopes [26]. In particular the name `parity polytope' suggests a connection to combinatorial optimization the electronic journal of combinatorics 27(3) (2020), #P3.16 https://doi.org/10.37236/8457 and polyhedral combinatorics; see [6, 10] for more. However, halfcubes also occur in algebraic/topological combinatorics [13, 14], convex algebraic geometry [22], and in many more areas. In this paper, we investigate basic properties of the following class of polytopes that contains cubes, simplices, cross-polytopes, and halfcubes. For a nonempty subset S of [0; d] := f0; 1; : : : ; dg, we define the S-hypersimplex d ∆(d; S) := conv v 2 f0; 1g : v1 + v2 + ··· + vd 2 S : In the context of combinatorial optimization these polytopes were studied by Gr¨otschel [15] associated to cardinality homogeneous set systems. Our name and notation derive from the fact that if S = fkg is a singleton, then ∆(d; S) =: ∆(d; k) is the well-known (d; k)-hypersimplex, the convex hull of all vectors v 2 f0; 1gd with exactly k entries equal to 1. This is a (d − 1)-dimensional polytope for 0 < k < d that makes prominent appearances in combinatorial optimization as well as in algebraic geometry [19]. We call S proper, if ∆(d; S) is a d-dimensional polytope, which, for d > 1, is precisely the case if jSj 6= 1 and S 6= f0; dg. For appropriate choices of S ⊆ [0; d], we get { the cube d = ∆(d; [0; d]), { the even halfcube Hd = ∆(d; [0; d] \ 2Z), { the simplex ∆d = ∆(d; f0; 1g), and { the cross-polytope ∆(d; f1; d − 1g) (up to linear isomorphism). In Section2, we study the vertices, edges, and facets of S-hypersimplices. Our study is guided by a nice decomposition of S-hypersimplices into Cayley polytopes of hypersimplices. In Section3 we return to the halfcube. A combinatorial d-cube has the interesting property that all pulling triangulations have the same number of d-dimensional simplices. The Freudenthal or staircase triangulation is a pulling triangulation and shows that the number of simplices is exactly d!. We show that the number of simplices in any pulling triangulation of Hd is independent of the order in which the vertices are pulled. More- over, we relate the full-dimensional simplices in any pulling triangulation of Hd to partial permutations and show that their number is given by d X d! t(d) = 2l−1 − l : l! l=3 For a polytope P ⊂ Rd and a linear function ` : Rd ! R, Billera and Sturmfels [4] associate the monotone path polytope Σ`(P ).This is a (dim P − 1)-dimensional poly- tope whose vertices parametrize all coherent `-monotone paths of P . As a particularly nice example, they show in [4, Example 5.4] that the monotone path polytope Σc(d), where c is the linear function c(x) = x1 + x2 + ··· + xd, is, up to homothety, the polytope Πd−1 = conv((σ(1); : : : ; σ(d)) : σ permutation of [d]) : For a point p 2 Rd, the convex hull of all permutations of p is called the permutahe- dron Π(p) and we refer to Πd−1 = Π(1; 2; : : : ; d) as the standard permutahedron. If the electronic journal of combinatorics 27(3) (2020), #P3.16 2 p has d distinct coordinates, then Π(p) is combinatorially (even normally) equivalent to Πd−1. For the case that p has repeated entries, these polytopes were studied by Billera- Sarangarajan [3] under the name of multipermutahedra. In Section4, we study maximal c-monotone paths in the vertex-edge-graph of ∆(d; S). We show that all c-monotone paths of ∆(d; S) are coherent and that essentially all multipermutahedra Π(p) for p 2 [0; d−1]d occur as monotone path polytopes of S-hypersimplices. We close with some questions and ideas regarding S-hypersimplices in Section5. 2 S-hypersimplices d The vertices of the d-cube can be identified with sets A ⊆ [d] and we write eA 2 f0; 1g for the point with (eA)i = 1 if and only if i 2 A. Let S ⊆ [0; d]. Since ∆(d; S) is a vertex-induced subpolytope of the cube, it is immediate that the vertices of ∆(d; S) are in bijection to [d] := fA ⊆ [d]: jAj 2 Sg : S P d This gives the number of vertices as jV (∆(d; S))j = s2S s . For a polytope P ⊂ Rd and a vector c 2 Rd, let c P := fx 2 P : hc; xi > hc; yi for all y 2 P g be the face in direction c. For example, unless S = f0g, ∆(d; S)ei is the convex hull [d] −ei of all eA with A 2 S with i 2 A. Likewise, unless S = fdg, ∆(d; S) = conv(eA : [d] ∼ d−1 A 2 S ; i 62 A). Under the identification fx : xi = 1g = R , this gives for jSj > 1 ∆(d; S)ei ∼= ∆(d − 1;S+) where S+ := fs − 1 : s 2 S; s > 0g ; (1) ∆(d; S)−ei ∼= ∆(d − 1;S−) where S− := fs : s 2 S; s < d − 1g : These faces will be helpful in determining the edges of ∆(d; S). For two sets A; B ⊆ [d], we denote the symmetric difference of A and B by A4B := (A [ B) n (A \ B). For two points p; q 2 Rd, we write [p; q] for the segment joining p to q. [d] Theorem 1. Let S = f0 6 s1 < ··· < sk 6 dg and A; B 2 S with jAj = si 6 sj = jBj. Then [eA; eB] is an edge of ∆(d; S) if and only if (i) A ⊂ B and j = i + 1, or (ii) i = j, jA4Bj = 2, and fsi − 1; si + 1g 6⊂ S. [d] Proof. Let A; B 2 S . If i 2 A \ B, then [eA; eB] is an edge of ∆(d; S) if and only if ei ei ∼ + ∼ [eA; eB] is an edge of ∆(d; S) . By (1), ∆(d; S) = ∆(d−1;S ) and [eA; eB] = [eAni; eBni]. Hence we can assume A \ B = ?. For i 2 [d] n (A [ B), we consider ∆(d; S)−ei and by the same argument we may also assume that A [ B = [d]. If A = ?, then B = [d] and [eA; eB] meets every ∆(d; k) in the relative interior for 0 < k < d. Hence [eA; eB] is an edge if and only if S = f0; dg, which gives us (i). the electronic journal of combinatorics 27(3) (2020), #P3.16 3 If 0 < si = jAj, then let i 2 A and j 2 B. Then [eA; eB] and [eA0 ; eB0 ] have the 0 0 same midpoint for A = (A n i) [ j and B = (B n j) [ i. Thus [eA; eB] is an edge of ∆(d; S) if and only if (A0;B0) = (B; A). This is the case precisely when jA4Bj = 2 and [d] A \ B; A [ B 62 S . Theorem1 makes the number of edges readily available. Corollary 2. The number of edges of ∆(d; S) is k X d − si d X sj(d − sj) d + ; s − s s 2 s i=1 i+1 i i j j where sk+1 := 0 and the second sum is over all 1 6 j 6 k, such that fsj − 1; sj + 1g 6⊂ S. Let us illustrate Theorem1 for the classical examples of S-hypersimplices. For the cube d = ∆(d; [0; d]) it states, that the edges are of the form [eA; eB] for any A ⊂ B ⊆ [d] such that jAj + 1 = jBj. For the halfcube Hd = ∆(d; [0; d] \ 2Z) we infer that there are d−3 d(d − 1)2 many edges for d > 3. As for the cross-polytope ∆(d; f1; d − 1g), every two vertices are connected by an edge, except for efig and e[d]nfig for all i 2 [d]. Theorem1 states that there are no long edges of ∆(d; S). We can make use of this fact to get a canonical decomposition of ∆(d; S).
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