S-hypersimplices, pulling triangulations, and monotone paths

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ 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.

The halfcubes H 1 and H 2 are a point and a segment, respectively, but for d ≥ 3, H d ⊂ R d is a fulldimensional 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 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 = {k} is a singleton, then ∆(d, S) =: ∆(d, k) is the well-known (d, k)-hypersimplex, the convex hull of all vectors v ∈ {0, 1} d 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 |S| = 1 and S = {0, d}. In Section 2, 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 Section 3 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 H d is independent of the order in which the vertices are pulled. Moreover, we relate the full-dimensional simplices in any pulling triangulation of H d to partial permutations and show that their number is given by For a polytope P ⊂ R d and a linear function : R d → 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 For a point p ∈ R d , the convex hull of all permutations of p is called the permutahedron Π(p) and we refer to Π d−1 = Π(1, 2, . . . , d) as the standard permutahedron. If p has d distinct coordinates, then Π(p) is combinatorially (even normally) equivalent to Π d−1 . For the case that p has repeated entries, these polytopes were studied by Billera-Sarangarajan [3] under the name of multipermutahedra. In Section 4, 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 ∈ [0, d − 1] d occur as monotone path polytopes of S-hypersimplices.
We close with some questions and ideas regarding S-hypersimplices in Section 5.
Acknowledgements. This paper grew out of a project that was part of the course Polytopes, Triangulations, 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.

S-hypersimplices
These faces will be helpful in determining the edges of ∆(d, S where we set s k+1 = 0 and the second sum is over all 1 ≤ j ≤ k, such that {s j − 1, s j + 1} ⊂ S.
We note the following consequence of Theorem 2.1.
Before we determine the facets of ∆(d, S), we recall some properties of permutahedra from [3] that we will also need in Section 4. A point p ∈ R d is decreasing if p 1 ≥ p 2 ≥ · · · ≥ p d . The permutahedron associated to p is the polytope Unless p i = p j for all i = j, Π(p) is a polytope of dimension d − 1 with affine hull given by H(p 1 + · · · + p d ).
Notice that Π(p) σu = σ −1 Π(p) u . Thus, if we want to determine the face Π(p) u up to permutation of coordinates, we can assume that u is decreasing. The Minkowski sum of two polytopes P, Q ⊂ R d is the polytope P + Q = {p + q : p ∈ P, q ∈ Q}.
Proposition 2.4. Let p, q ∈ R d be decreasing. Then Proof. Set P := Π(p)+Π(q). Clearly σ(p+q) = σp+σq for all permutations σ and therefore every vertex of Π(p + q) is a vertex of P . For the converse, let c be such that P c = {v} is a vertex. Since P is invariant under coordinate permutations, we can assume that c is decreasing. Furthermore (Π(p) + Π(q)) c = Π(p) c + Π(q) c and it follows that v = p + q. Hence, every vertex of P is of the form σ(p + q) for some permutation σ, which completes the proof.
The facets of permutahedra were described by Billera-Sarangarajan [3]. We recall their characterization.
In order to determine the facets of ∆(d, S), we appeal to the decomposition (2).
is any other facet, then its vertices cannot have all the same cardinality.
Hence, as a first step, we determine the facets of ∆(d, s i , s i+1 ) that are not equal to ∆(d, s i ) and ∆(d, s i+1 ).
Let S = {k < l} be proper. An easy calculation shows that is a facet of the right-hand side and every facet arises that way. Hence it suffices to determine the facets of ∆(d, k, l) := ∆(d, k) + ∆(d, l). We will need the notion of a join of two polytopes: If P, Q ⊂ R d are polytopes such that their affine hulls are skew, i.e., nonparallel and disjoint, then P * Q := conv(P ∪ Q) is called the join of P and Q. Every Proposition 2.6. Let 1 ≤ k < l < d. In addition to the facets ∆(d, k, l) 1 = ∆(d, l) and ∆(d, k, l) −1 = ∆(d, k), there are Proof. We first determine the facets of ∆(d, k, l). Using Proposition 2.4, we see that ∆(d, k, l) is the permutahedron Π(2 k , 1 l−k , 0 d−l ). Theorem 2.5 yields that the facet directions of ∆(d, k, l) are given c = αe I + βe I c for ∅ = I ⊂ [d] with |I| = 1, |I| = d − 1, or k < |I| < l and α > β. In particular, for every I there is, up to scaling, a unique choice for α and β so that ∆(d, k, l) c is a facet.
This also shows that the given subspaces are skew and, since they lie in H(k) and H(l) respectively, are disjoint. This shows that ∆(d, l, k) ∼ = ∆(h, k) * ∆(d − h, l − h).
It follows from Proposition 2.6 that ∆(d, k, l) and ∆(d, l, m) for 0 < k < l < m < d never have facet normals of type (v) in common. This gives us the following description of facets of S-hypersimplices; see also [15].
Theorem 2.7. Let S = {0 ≤ s 1 < · · · < s k ≤ d} be proper. Then ∆(d, S) has the following facets Proof. By decomposition (2), every facet F of ∆(d, S) determines a facet of ∆(d, s i , s i+1 ) for some 1 ≤ i < k and F is decomposed by this collection of facets. By examining the possible facet normals of ∆(d, s i , s i+1 ), the statement readily follows. The two facets of type (i), (ii), and those of type (iii) and (iv) are simplices. As for type (v) this is a join of two simplices and thus also a simplex.
The description of combinatorial type of each facet also leads to the number of k-dimensional faces for 0 ≤ k < d; cf. [21].

Pulling triangulations
A subdivision S of a d-dimensional polytope P ⊂ R d is a collection S = {P 1 , . . . , P m } of d-polytopes such that P = P 1 ∪ · · · ∪ P m and P i ∩ P j is a face of P i and P j for all 1 ≤ i < j ≤ m. If all polytopes P i are simplices, then S is called a triangulation. Triangulations are the method-of-choice for various computations on polytopes including volume, lattice point counting, or, more generally, computing valuations; see [8].
A powerful method for computing a triangulation is the so-called pulling triangulation. Let P be a d-polytope and v ∈ V (P ) a vertex. Let F 1 , . . . , F m be the facets of P not containing v. A key insight is that the collection of polytopes constitutes a subdivision of P . This idea can be extended to obtain triangulations. Let be a partial order on the vertex set V (P ) such that every nonempty face F ⊆ P has a unique minimal element with respect to . We denote the minimal vertex of F by v F . The pulling triangulation Pull (P ) of P is recursively defined as follows. If P is a simplex, then Pull (P ) = {P }. Otherwise, we define where the union is over all facets F ⊂ P that do not contain v P and where v P * Pull (F ) := {v P * Q : For the cube d , or more generally the class of compressed polytopes [25], it can be shown that every simplex S in a pulling triangulation of d has the same volume 1 d! . Thus, every pulling triangulation has exactly d! many simplices, independent of the chosen order .
Recall that the halfcube is the S-hypersimplex H d = ∆(d, [0, d] ∩ 2Z). For d ≥ 5 it is not true that the simplices in a pulling triangulation of H d all have the same volume. The main result of this section is that still the number of simplices in a pulling triangulation is independent of the choice of .
The proof of Theorem 3.1 is in two parts. We first show that the number of simplices of Pull (H d ) is independent of . This yields a recurrence relation on t(d). In the second part we review the construction of Pull (H d ) from the perspective of choosing facets, which yields a combinatorial interpretation for t(d) and which then verifies the stated expression.
From Theorem 2.7 we infer the following description of facets of H d for d ≥ 3: For every i = 1, . . . , d we have Proof. We prove the result by induction on d.
where the last equality follows by induction.
Let P ⊂ R d be a full-dimensional polytope with suitable partial order on V (P ). Every simplex in Pull (P ) corresponds to a chain of faces The corresponding simplex is then given by v G 0 * v G 1 * · · · * G k . If P is a simple polytope with facets F 1 , . . . , F m , then any such chain of faces is given by an ordered sequence of distinct indices h 1 , h 2 , . . . , h k such that For the d-dimensional cube d , the facets can be described by Observe that for any vertex v ∈ d and i ∈ [d], we have that v ∈ K 0 i or v ∈ K 1 i . This means that for any permutation σ = i 1 i 2 · · · i d of [d] there are δ 1 , δ 2 , . . . , δ d−1 ∈ {0, 1} such that (i 1 , δ 1 ), . . . , (i d−1 , δ d−1 ) come from a simplex in Pull ( d ). This shows that |Pull ( d )| = d! independent of the order .
We call a sequence τ = i 1 i 2 . . . i k with i 1 , . . . , i k ∈ [d] a partial permutation if i s = i t for s = t. We simply write [d] \ τ for [d] \ {i 1 , . . . , i k }. The following Proposition completes the proof of Theorem 3.1.  is not a simple polytope. However, it follows from Theorem 2.7 that the faces of H d are halfcubes or simplices. If G ⊂ H d is a face linearly isomorphic to a halfcube of dimension d − k ≥ 4, then G is a simple face in the sense that G is precisely the intersection of k halfcube facets. Every chain of faces (4) corresponds to some (i 1 , δ 1 ), . . . , and G k is a simplex facet of G k−1 not containing v G k−1 . This gives rise to a unique partial permutation τ = i 1 i 2 . . . i k−1 . To see that any such partial permutation can arise, we observe that again with H d−k+1 embedded in {x : x i 1 = · · · = x i k−1 = 0} and v G k−1 = 0. Now any simplex facet of H d−k+1 corresponds to an odd-cardinality subset B ⊂ [d] \ τ with |B| = 1.

Monotone paths
Let P ⊂ R d be a polytope and : R d → R a linear function. An -monotone path of P is a sequence of vertices ] is an edge of P for i = 1, . . . , k − 1 and More generally, a collection of faces F 1 , F 2 , . . . , F k of P is an induced subdivision of the segment (P ) if F − 1 and F k is a face of P − and P , respectively, and If is generic, that is, if is not constant on edges of P , then the minimum/maximum of on every nonempty face F is attained at a unique vertex. In this case F ± i is a vertex for all i and a induced subdivision is called a cellular string. An induced subdivision F 1 , . . . , F h is a refinement if for every 1 ≤ i ≤ k, there are 1 ≤ s < t ≤ h such that F s , . . . , F t is a induced subdivision of (F i ).
The collection of all induced subdivisions of (P ) is partially ordered by refinement and is called the Baues poset of (P, ). The minimal elements in the Baues poset are exactly the -monotone paths. Monotone paths are quintessential in the study of simplex-type algorithms in linear programming but they are also studied in topology in connection with iterated loop spaces; see [2,20]. For the linear function c(x) = x 1 + · · · + x d , Corollary 2.2 readily yields the c-monotone paths of ∆(d, S).
Corollary 4.1. Let S = {s 1 < s 2 < · · · < s k } be proper. The c-monotone paths correspond to sequences A -monotone path W is coherent if W is a monotone path with respect to the shadow-vertex algorithm; see [5,17]. That is, if there is linear function h W : R d → R such that under the projection π : R d → R 2 given by π(x) = ( (x), h W (x)), the path W is mapped to one of the two paths in the boundary of the polygon π(P ). Figure 1 shows that in general coherent paths constitute a proper subset of all -monotone paths and it is interesting to determine for which pairs (P, ) all -monotone paths are coherent; see, for example, the recent paper [9]. The S-hypersimplices with the linear function c(x) are examples of this.  Proof. Let A 1 ⊂ A 2 ⊂ · · · ⊂ A k be a c-monotone path. For the linear function S is maximal if and only if B ∈ {A 1 , . . . , A k }.
The monotone path polytope Σ (P ) is a convex polytope of dimension dim P − 1 whose face lattice is isomorphic to the poset of coherent subdivisions. The construction is a special case of fiber polytopes of Billera and Sturmfels [4]. Let (P ) = [a, b] ⊂ R. A section of (P, ) is a continuous function γ : [a, b] → P such that (γ(t)) = t for all a ≤ t ≤ b. Following [4], the monotone path polytope is defined as We now determine the monotone path polytopes of ∆(d, S) with respect to the natural linear function c(x) = x 1 + · · · + x d . Let us first observe that for S ⊂ [d − 1] the c-monotone paths of ∆(d, S) and ∆(d, S ∪ {0, d}) are in bijection. Clearly every c-monotone path of ∆(d, S ∪ {0, d}) restricts to a cmonotone path of ∆(d, S). Conversely, if A 1 ⊂ · · · ⊂ A k corresponds to a c-monotone path, then ∅ =: is the unique extension to a c-monotone path of ∆(d, S ∪ {0, d}).

Further questions
Volumes and Gröbner bases. Laplace and later Stanley [24] showed that the volume of ∆(d, i, i + 1) where A(d, i) counts the number of permutations σ of [d] with i descents, that is, the number of 1 ≤ i < d such that σ(i) > σ(i + 1); see also [18,23]. This implies that d! vol ∆(d, [k, l]) is the number of permutations of [d] with descent number in [k, l] = {k, k + 1, . . . , l} for any k < l. It would be very interesting to know if vol ∆(d, S) has a combinatorial interpretation for all S. In light of (2) it would be sufficient to determine vol ∆(d, k, l) for l − k > 1.
For 0 ≤ k < d, the hypersimplices ∆(d, k, k + 1) ∼ = ∆(d, k + 1) are alcoved polytopes in the sense of Lam-Postnikov [18] and hence come with a canonical square-free and unimodular triangulation. This is reflected by the fact that the associated toric ideals have quadratic and square-free Gröbner bases with respect to the reverse-lexicographic term order.
For general k < l, the polytopes ∆(d, k, l) are not alcoved anymore. It would be interesting if ∆(d, k, l) has a unimodular triangulation or square-free Gröbner basis. 5.1. Extension complexity. An extension of a polytope P is a polytope Q together with a surjective linear projection Q → P . The extension complexity ext(P ) of P is the minimal number of facets of an extension of P . This is a parameter that is of interest in combinatorial optimization [16]. It was shown in [12] that ext(∆(d, k, k + 1)) = 2d for 1 ≤ k ≤ d − 2.
A realization of the join of two polytopes P, Q ⊂ R d is given by P * Q = conv((P × 0 × 0) ∪ (0 × Q × 1)). If P and Q has m and n facets, respectively, then P * Q has m + n facets. Balas' union bound [1] is the observation that P * Q → P ∪ Q and hence ext(P ∪ Q) ≤ ext(P ) + ext(Q). Iterating the join over the pieces of the decomposition 2 shows the following. This is a nontrivial bound as the number of facets of ∆(d, S) is at least 2 + 2d + r ∈S d r . To illustrate, note that the number of facets of the halfcube H d for d ≥ 5 is 2d + 2 d−1 whereas the bounded afforded by Proposition 5.1 is ≤ d 2 . Carr and Konjevod [6] gave an extension of H d of size linear in d. It would be interesting to know lower bounds on the extension complexity of ∆(d, S), maybe using the approach via rectangular covering; c.f. [12].