S-HYPERSIMPLICES, PULLING TRIANGULATIONS, AND MONOTONE PATHS SEBASTIAN MANECKE, RAMAN SANYAL, AND JEONGHOON SO 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, crosspoly- topes, 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. 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 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 Grötschel [15] associated to cardinality homogeneous set systems. Our name and notation derive from the fact that if S = fkg is a arXiv:1812.07491v2 [math.CO] 28 Nov 2019 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). Date: December 2, 2019. 2010 Mathematics Subject Classification. 52B20, 52B12. Key words and phrases. hypersimplex, pulling triangulation, permutahedra, monotone path polytope. 1 2 SEBASTIAN MANECKE, RAMAN SANYAL, AND JEONGHOON SO 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. Moreover, 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 d d For a polytope P ⊂ R and a linear function ` : R ! R, Billera and Sturmfels [4] associate the monotone path polytope Σ`(P ).This is a (dim P − 1)-dimensional polytope 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]) : d For a point p 2 R , the convex hull of all permutations of p is called the permutahedron Π(p) and we refer to Πd−1 = Π(1; 2; : : : ; d) as the standard permutahedron. If 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. Acknowledgements. This paper grew out of a project that was part of the course Polytopes, Triangula- tions, and Applications at Goethe University Frankfurt in spring 2018. We thank Anastasia Karathanasis for her support in the early stages of this project. We also thank Jesús de Loera, Georg Loho, and the anonymous referee for many helpful remarks. 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 . d d For a polytope P ⊂ R and a vector c 2 R , let P c := fx 2 P : hc; xi ≥ hc; yi for all y 2 P g e be the face in direction c. For example, unless S = f0g, ∆(d; S) i is the convex hull of all eA with [d] −ei [d] A 2 S with i 2 A. Likewise, unless S = fdg, ∆(d; S) = conv(eA : A 2 A 2 S ; i 62 A). Under the ∼ d−1 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 : S-HYPERSIMPLICES, PULLING TRIANGULATIONS, AND MONOTONE PATHS 3 These faces will be helpful in determining the edges of ∆(d; S). For two sets A; B ⊆ [d], we denote the d symmetric difference of A and B by A4B := (A [ B) n (A \ B). For two points p; q 2 R , we write [p; q] for the segment joining p to q. [d] Theorem 2.1. Let S = f0 ≤ s1 < ··· < sk ≤ dg and A; B 2 S with jAj = si ≤ 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 [eA; eB] is an ei ei ∼ + ∼ edge of ∆(d; S) . By (1), ∆(d; S) = ∆(d − 1;S ) and [eA; eB] = [eAni; eBni]. Hence we can assume −e A \ B = ?. For i 2 [d] n (A [ B), we consider ∆(d; S) i 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). If 0 < si = jAj, then let i 2 A and j 2 B. Then [eA; eB] and [eA0 ; eB0 ] have the same midpoint for 0 0 0 0 A = (A n i) [ j and B = (B n j) [ i. Thus [eA; eB] is an edge of ∆(d; S) if and only if (A ;B ) = (B; A). [d] This is the case precisely when jA4Bj = 2 and A \ B; A [ B 62 S . Theorem 2.1 makes the number of edges readily available. Corollary 2.2. 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 we set sk+1 = 0 and the second sum is over all 1 ≤ j ≤ k, such that fsj − 1; sj + 1g 6⊂ S. Let us illustrate Theorem 2.1 for the classical examples of S-hypersimplices. For 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 d−3 Hd = ∆(d; [0; d] \ 2Z) we infer that there are 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]. Theorem 2.1 states that there are no long edges of ∆(d; S).