Volume and lattice points counting for the cyclopermutohedron
aa r X i v : . [ m a t h . M G ] M a y VOLUME AND LATTICE POINTS COUNTING FORTHE CYCLOPERMUTOHEDRON
ILYA NEKRASOV, GAIANE PANINA
Abstract.
The face lattice of the permutohedron realizes thecombinatorics of linearly ordered partitions of the set [ n ] = { , ..., n } .Similarly, the cyclopermutohedron is a virtual polytope that real-izes the combinatorics of cyclically ordered partitions of [ n ].It is known that the volume of the standard permutohedronequals the number of trees with n labeled vertices multiplied by √ n . The number of integer points of the standard permutohedronequals the number of forests on n labeled vertices.In the paper we prove that the volume of the cyclopermutohe-dron also equals some weighted number of forests, which eventu-ally reduces to zero. We also derive a combinatorial formula forthe number of integer points in the cyclopermutohedron.Another object of the paper is the configuration space of a polyg-onal linkage L . It has a cell decomposition K ( L ) related to the facelattice of cyclopermutohedron. Using this relationship, we intro-duce and compute the volume V ol ( K ( L )). Introduction
The standard permutohedron Π n is defined (see [11]) as the convexhull of all points in R n that are obtained by permuting the coordinatesof the point (1 , , ..., n ). It has the following properties:(1) Π n is an ( n − k -faces of Π n are labeled by ordered partitions of the set[ n ] = { , , ..., n } into ( n − k ) non-empty parts.(3) A face F of Π n is contained in a face F ′ iff the label of F refinesthe label of F ′ .Here and in the sequel, we mean the order-preserving refine-ment. For instance, the label ( { , } , { , } , { } , { } ) refines thelabel ( { , } , { , } , { , } ), but does not refine ( { , } , { , } , { , } ).(4) The permutohedron is a zonotope , that is, Minkowski sum ofline segments q ij , whose defining vectors are { e i − e j } i
2. The incidence relation in CP n +1 corresponds to the refinement: a cell F contains a cell F ′ whenever thelabel of F ′ refines the label of F .The cyclopermutohedron is defined explicitly, as a weighted Minkowskisum of line segments.In the paper we prove that the volume of the cyclopermutohedronequals some weighted number of forests. Making use of the theory ofAbel polynomials, we eventually reduce the expression to zero. We alsogive a combinatorial formula for the number of integer points in thecyclopermutohedron.Another object of the paper is the configuration space, or modulispaces of a polygonal linkage L . One of the motivations for introducingthe cyclopermutohedron is that CP n +1 is a ”universal” polytope formoduli spaces of polygonal linkages. Namely, given a flexible polygon L , the space of its planar shapes (that is, the configuration space)has a cell decomposition K ( L ), whose combinatorics embeds in thecombinatorics of the face poset of cyclopermutohedron. Using thisrelationship we introduce and compute the volume V ol ( K ( L )).The paper is organized as follows. In Section 2 we give all necessaryinformation about virtual polytopes, and also the definition and prop-erties of the cyclopermutohedron. Abel polynomials are also sketchedin the section.In Section 3 we explain the meaning of the ”volume of cyclopermu-tohedron”, and prove that it equals zero. In Section 4 we explain therelationship with polygonal linkages and give a formula for the volumeof the configuration space (Theorem 3). OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 3
Finally, in Section 5 we compute the number of integer points in thecyclopermutohedron (Theorem 4).
Acknowledgements.
The present research is supported by RFBR,research project No. 15-01-02021. The first author was also sup-ported by the Chebyshev Laboratory under RF Government grant11.G34.31.0026, and JSC ”Gazprom Neft”.2.
Theoretical backgrounds
Virtual polytopes.
Virtual polytopes appeared in the literatureas useful geometrization of Minkowski differences of convex polytopes.A detailed discussion can be found in [4, 6, 8]; below we give just abrief sketch. As a matter of fact, in the paper (except for Section 4) weneed no geometrization. Even for volume and integer point counting,it is sufficient to know that virtual polytopes form the Grothendieckgroup associated to the semigroup of convex polytopes.More precisely, a convex polytope is the convex hull of a finite, non-empty point set in the Euclidean space R n . Degenerate polytopes arealso included, so a closed segment and a point are polytopes, but notthe empty set. We denote by P + the set of all convex polytopes.Let K and L ∈ P + be two convex polytopes. Their Minkowski sum K + L is defined by: K + L = { x + y : x ∈ K, y ∈ L } . Minkowski addition turnes the set P + to a commutative semigroupwhose unit element is the convex set containing exactly one point E = { } . Definition 1.
The group P of virtual polytopes is the Grothendickgroup associated to the semigroup P + of convex polytopes under Minkowskiaddition.The elements of P are called virtual polytopes . More instructively, P can be explained as follows.(1) A virtual polytope is a formal difference K − L .(2) Two such expressions K − L and K − L are identified, when-ever K + L = K + L .(3) The group operation is defined by( K − L ) + ( K − L ) := ( K + K ) − ( L + L ) . It is important that the notions of ”volume” and ”number of integerpoints” extend nicely to virtual polytope. We explain these construc-tions in the subsequent sections.
ILYA NEKRASOV, GAIANE PANINA
Cyclopermutohedron. [7]Assuming that { e i } ni =1 are standard basic vectors in R n , define thepoints R i = P ni =1 ( e j − e i ) = ( − , ... − , n − , − , ... − , − , − , ) ∈ R n ,i and the following two families of line segments: q ij = [ e i , e j ] , i < j and r i = [0 , R i ] . We also need the point e = (1 , , ..., ∈ R n .The cyclopermutohedron is a virtual polytope defined as the Minkowskisum: CP n +1 := M i The Minkowski sum M i 2, the k -dimensional faces of CP n +1 are labeledby (all possible) cyclically ordered partitions of the set [ n + 1]into ( n − k + 1) non-empty parts.(2) A face F ′ is a face of F whenever the label of F ′ refines thelabel of F . Here we mean order preserving refinement. OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 5 Abel polynomial and rooted forests. [10]A rooted forest is a graph equal to a disjoint union of trees, whereeach of the trees has a marked vertex.The Abel polynomials form a sequence of polynomials, where the n -th term is defined by A n,a ( x ) = x ( x − an ) n − . A special case of the Abel polynomials with a = − A n ( x ) := A n, − ( x ) = x ( x + n ) n − is the n -thAbel polynomial, then A n ( x ) = n X k =0 t n,k · x k , where t n,k is the number of forests on n labeled vertices consisting of k rooted trees.3. Volume of cyclopermutohedron equals zero The notion of volume extends nicely from convex polytopes to virtualpolytopes. We explain below the meaning of the volume of a virtualzonotope .Assume we have a convex zonotope Z ⊂ R n , that is, the Minkowskisum of some linear segments { s i } mi =1 : Z = m M i =1 s i . For each subset I ⊂ [ m ] such that | I | = n , denote by Z I the el-ementary parallelepiped , or the brick spanned by n segments { s i } i ∈ I ,provided that the defining vectors of the segments are linearly inde-pendent. In other words, the brick equals the Minkowski sum Z I = M I s i . It is known that Z can be partitioned into the union of all such Z I ,which implies immediately V ol ( Z ) = X I ⊂ [ m ] , | I | = n V ol ( Z I ) = X I ⊂ [ m ] , | I | = n | Det ( S I ) | , where S I is the matrix composed of defining vectors of the segmentsfrom I .Now take positive λ , ..., λ m and sum up the dilated segments λ i s i .Clearly, we have ILYA NEKRASOV, GAIANE PANINA V ol (cid:16) m M i =1 λ i s i (cid:17) = X I ⊂ [ m ] , | I | = n Y i ∈ I λ i · | Det ( S I ) | . For fixed s i , we get a polynomial in λ i , which counts not only thevolume of convex zonotope (which originates from positive λ i ), butalso the volume of a virtual zonotope, which originates from any real λ i , including negative ones, see [4, 8].So, one can use the above formula as the definition of the volume ofa virtual zonotope.An almost immediate consequence is: Lemma 1. Let E = E n be the set of edges of the complete graph K n .The ( n − -volume of the cyclopermutohedron can be computed by the for-mula: V ol ( CP n +1 ) = V ol (cid:16) M i 1. That is, we deal with ( n − n -volume by adding e = (1 , , ..., , , ) and divid-ing by | e | = √ n . (cid:3) Remark 2. The formula for the volume of a virtual zonotope also has ageometrical meaning which we briefly sketch here. Due to Brianchon-Gram decomposition of virtual polytopes (see [8] or [4] ), any virtualpolytope can be viewed as a codimension one homological cycle, andtherefore possesses a well-defined (algebraic) volume.For a virtual zonotope, the associated cycle decomposes into homolog-ical sum of elementary bricks, but the latter should be understood alsoas homological cycles coming with different orientations. More pre-cisely, if the number of negative λ i in the sum L ni =1 λ i s i is even, thenthe corresponding elementary brick equals the boundary of elementary OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 7 parallelepiped ∂ (cid:16) L ni =1 | λ i | s i (cid:17) with the positive orientation (that is,cooriented by the outer normal vector). If the number of negative λ i isodd, we have the same cycle with the negative orientation. Theorem 1. V ol ( CP n +1 ) = 0 . Proof. Keeping in mind Lemma 1, let’s first fix I and M with | I | + | M | = n − 1, and compute one single summand | Det ( q ij , r k , e ) | ( ij ) ∈ I, k ∈ M .If M = ∅ , the determinant equals 1 iff the set I gives a tree. Oth-erwise it is zero. (This is the reason for the volume formula of thepermutohedron.)Assume now that M is not empty. | Det ( q ij , r k , e ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · − · · · − · · · − · · · − · · · − · · · − · · · 11 0 · · · − · · · − · · · 10 0 · · · n − · · · − · · · 10 1 · · · − · · · − · · · 10 0 · · · − · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =Adding e to all the columns r i , we get:= n | M | · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · · · · · · · − · · · · · · − · · · · · · 11 0 · · · · · · · · · 10 0 · · · · · · · · · 10 1 · · · · · · · · · 10 0 · · · · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n | M | · ( ∗ ) . ILYA NEKRASOV, GAIANE PANINA We wish to proceed in a similar way, that is, add the columns containingthe unique entry 1 to other columns chosen in an appropriate way. Toexplain this reduction let us give two technical definition. Definition 2. A decorated forest F = ( G, M ) is a graph G = ([ n ] , I ) without cycles on n labeled vertices together with a set of marked ver-tices M ⊂ [ n ] such that the following conditions hold: (1) Number of marked vertices | M | + number of edges | I | equals n − . (2) Each connected component of G has at most one marked vertex. Immediate observations are:Each decorated forest has exactly one connected component with novertices marked. We call it a free tree . Denote by N ( F ) the number ofvertices of the free tree.Each decorated forest is a disjoint union of the free tree and somerooted forest. The number of rooted trees equals | M | .Each decorated forest F yields a collection of { e ij , r k } ( ij ) ∈ I, k ∈ M ,whose above determinant ( ∗ ) we denote by | Det ( F ) | for short.For instance, for the first decorated forest in Figure 1, we have N ( F ) = 2 , | M | = 1 . Now we define the reduction of a decorated forest (see Figure 1 forexample). It goes as follows.Assume we have a decorated forest. Take a marked vertex i and anincident edge ( ij ). Remove the edge and mark the vertex j . Repeatuntil is possible.Roughly speaking, a marked vertex i kills the edge ( ij ) and generatesa new marked vertex j . Figure 1. Reduction process for a forest with N ( F ) =2 , | M | ( F ) = 1. Grey balls denote the marked vertices.An obvious observation is: Lemma 2. (1) The free tree does not change during the reduction. OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 9 (2) The reduction brings us to a decorated forest with a unique freetree. All other trees are one-vertex trees, and all these verticesare marked. (3) The reduction can be shortened: take the connected componentsone by one and do the following. (a) If a connected component has no marked vertices, leave itas it is. (b) If a connected component has a marked vertex, eliminateall its edges and mark all its vertices. (4) The reduction does not depend on the order of the marked ver-tices we deal with. (cid:3) Before we proceed with the proof of Theorem 1, prove the lemma: Lemma 3. (1) For each decorated forest F , | Det ( F ) | = N ( F ) . (2) If a collection { e ij , r k } does not come from a decorated forest,that is, violates condition (2) from Definition 2, then | Det ( e ij , r k ) | = 0 . Proof of the lemma. (1) For a decorated forest, we manipulate withthe columns according to the reduction process. We arrive at a matrixwhich (up to a permutation of the columns and up to a sign) is: (cid:18) A O O E (cid:19) . Here A is the matrix corresponding to the free tree, E is the unitmatrix, and the very last column is e . Its determinant equals 1.(2) If the collection of vectors does not yield a decorated forest, thatis, there are two marked vertices on one connected component, theanalogous reduction gives a zero column. (cid:3) Basing on Lemmata 3 and 1, we conclude: V ol ( CP n +1 ) = 1 √ n X F ( − n ) | M ( F ) | · N ( F ) = , where the sum extends over all decorated forests F on n vertices.(Remind that M ( F ) is the set of marked vertices, N ( F ) is the numberof vertices of the free tree.)Next, we group the forests by the number N = N ( F ) and write = 1 √ n n X N =1 (cid:18) nN (cid:19) N N − · N X f ( − n ) C ( f ) == 1 √ n n X N =1 (cid:18) nN (cid:19) N N − X f ( − n ) C ( f ) = 1 √ n · ( ∗∗ ) , where the second sum ranges over all rooted forest on ( n − N ) labeledvertices, C ( · ) is the number of connected components.Let us explain this in more details.(1) N ranges from 1 to n . We choose N vertices in (cid:0) nN (cid:1) differentways and place a tree on these vertices in N N − ways.(2) On the rest of the vertices we place a rooted forest.Recalling that t n − N,k the number of forests on ( n − N ) labeled verticesof k rooted trees, we write:( ∗∗ ) = n X N =1 (cid:18) nN (cid:19) N N − n − N X k =1 ( − n ) k · t n − N,k . Section 2.3 gives us: n X k =0 t n,k x k = A n ( x ) , where A n ( x ) = x ( x + n ) n − is the Abel polynomial.Setting − n = x , we get n − N X k =1 ( − n ) k · t n − N,k = A n − N ( − n ) . Thus ( ∗∗ ) converts to n X N =1 (cid:18) nN (cid:19) N N − A n − N ( − n ) =: Q n . Applying the definition of A n − N ( − n ), we get Q n = n X N =1 (cid:18) nN (cid:19) N N − ( − n )( − n + n − N ) n − N − = ( − n n · n X N =1 ( − N (cid:18) nN (cid:19) N n − . Introduce the following polynomial: p ( x ) := n X N =0 N n − (cid:18) nN (cid:19) x N , OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 11 for which we have Q n = p ( − p ( x ) := (1 + x ) n = n X N =0 (cid:18) nN (cid:19) x N ,p i ( x ) := x · p ′ i − ( x ) = n X N =0 N i (cid:18) nN (cid:19) x N . We clearly have p ( x ) = p n − ( x ). Besides, (1 + x ) n − k divides p k ( x ),which implies Q n = 0. (cid:3) Polygonal linkages: volume of the configuration space Definitions and notation. A flexible ( n +1) -polygon , or a polyg-onal ( n +1) -linkage is a sequence of positive numbers L = ( l , . . . , l n +1 ).It should be interpreted as a collection of rigid bars of lengths l i joinedconsecutively in a closed chain by revolving joints. We always assumethat the triangle inequality holds, that is, ∀ j, l j < n +1 X i =1 l i which guarantees that the chain of bars can close.We also assume that the last bar is the longest one: ∀ j l n +1 ≥ l j . A planar configuration of L is a sequence of points P = ( p , . . . , p n +1 ) , p i ∈ R with l i = | p i p i +1 | , and l n +1 = | p n +1 p | . As follows from the definition,a configuration may have self-intersections and/or self-overlappings. The moduli space, or the configuration space M ( L ) is the space ofall configurations modulo orientation preserving isometries of R .Equivalently, we can define M ( L ) as M ( L ) = { ( u , ..., u n +1 ) ∈ ( S ) n +1 : n +1 X i =1 l i u i = 0 } /SO (3) . The (second) definition shows that M ( L ) does not depend on theordering of { l , ..., l n +1 } ; however, it does depend on the values of l i .Let us comment on this dependance. Consider ( l , ..., l n +1 ) as a pointin the parameter space R n +1 . The hyperplanes in R n +1 defined by all possible equations n +1 X i =1 ε i l i = 0 with ε i = ± walls . Throughout the section we assume that the point { l , ..., l n +1 } belongs to none of the walls. This genericity assumptionimplies that the moduli space M ( L ) is a closed ( n − R n +1 into a number of chambers ; the topologyof M ( l , ..., l n +1 ) depends only on the chamber containing { l , ..., l n +1 } (see [1]).The manifold M ( L ) is already well studied. In this paper we makeuse of the described below cell structure on the space M ( L ).4.2. The complex K ( L ) . Assume that ( l , ..., l n +1 ) is fixed.A set I ⊂ [ n + 1] = { , , ..., n + 1 } is called short , if X i ∈ I l i < n +1 X i =1 l i . Otherwise I is a long set.A partition of the set [ n + 1] is called admissible if all the sets in thepartition are short. Theorem 2. [6] There is a structure of a regular CW-complex K ( L ) onthe moduli space M ( L ) . Its complete combinatorial description readsas follows: (1) k -cells of the complex K ( L ) are labeled by cyclically ordered ad-missible partitions of the set [ n + 1] into ( n + 1 − k ) non-emptyparts. (2) A closed cell C belongs to the boundary of some other closedcell C ′ iff the partition λ ( C ) is finer than λ ( C ′ ) . A remark on notation. We write a cyclically ordered partition asa (linearly ordered) string of sets where the set containing the entry” n ” stands on the last position.We stress that the order of the sets matters, whereas there is noordering inside a set. For example,( { }{ }{ , , , } ) = ( { }{ }{ , , , } ) = ( { }{ }{ , , , } ) . Example 1. Assume that l n +1 = n X i =1 l i − ε, OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 13 where ε is small. In this case the moduli space M ( L ) is the sphere S ( n − , see [1] , and the complex K ( L ) is isomorphic to the boundarycomplex of the permutohedron Π n . For any ( n + 1)-linkage L , the complex K ( L ) automatically embedsin the face complex of cyclopemutohedron CP n +1 , and therefore can berealized by a polyhedron which we denote by P ( L ). Vividly speaking,the polyhedron P ( L ) is patched of those faces of the cyclopermutohe-dron, whose labels are admissible partitions. Example. For L as in Example 1, P ( L ) equals the boundary of the permuto-hedron Π n . Example. Let n = 5, L = (1 . , , , . , . P ( L ) is the cylinder de-picted in Fig. 2. The two shadowed faces are labeled by ( { }{ } ) and( { }{ } ). Since the partitions ( { }{ }{ } ) and ( { }{ }{ } ) arenon-admissible, these faces of permutohedron are removed, whereas allother faces of the permutohedron survive. There are also six ”diagonal”rectangular faces. They are labeled by ( { }{ }{ } ), ( { , }{ }{ } ),( { }{ }{ } ), ( { }{ }{ } ), ( { }{ }{ } ), and ( { }{ }{ } ). Figure 2. The complex K ( L ) for the 5-linkage L = (1 , 2; 1; 1; 0 , 8; 2 , Volume of the complex K ( L ) . Following the ideology of Re-mark 2, P ( L ) can be viewed as a codimension one homological cycle(or as a generalization of closed piecewise linear oriented manifold) in the Euclidean space. Therefore it makes sense to speak of the vol-ume of the part of the space bounded by P ( L ). Since P ( L ) may havemany self-intersections, the volume means the algebraic volume , thatis, multiplicities (which can be also negative) are taken into account.Let us explain this in more details. For each point x ∈ R n , denoteby ind x ( P ( L )) the index of the cycle with respect to the point x . Thenby the volume of the configuration space we mean V ol ( M ( L )) := V ol ( P ( L )) := Z R n ind x ( P ( L )) dx. Definition 3. For an ( n + 1) -linkage L , a decorated forest F on n labeled vertices is called non-admissible, if the vertex set of the free treeis a long set. In notation of Section 3, the following lemma holds: Lemma 4. For a ( n + 1) -linkage L , we have: V ol ( M ( L )) = 1 √ n X non − admissible F ( − n ) | M ( F ) | · N ( F ) , where the sum ranges over all non-admissible decorated forests on n labeled vertices.We remind that | M ( F ) | denotes the number of marked vertices, N ( F ) is the number of vertices of the free tree. Proof. Let us take the linkage L = ( l , ..., l n , λ ) assuming that thevalue of λ continuously and monotonly changes from n X i =1 l i − ε to l n +1 . In the beginning we have the permutohedron Π n , whose volume wealready know. At the end, we have P ( L ), whose volume we wish tocalculate. In between we have a (finite) number of Morse surgeries,and we can control the behavior of the volume at each of the surgeries.Prove first that the formula holds true for λ = P ni =1 l i − ε . Indeed,for this particular λ , a ”decorated non-admissible forest on n vertices”means just ”a free tree on n vertices”, so the statement of the theoremreduces to the formula for the volume of the standard permutohedron,see Section 1.Now we start changing λ . This means that we have a path in theparameter space R n +1 , which crosses some of the walls. We can assumethat the walls are crossed one by one; if this is not the case, we perturbgenerically the original lengths l i . OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 15 Once we cross a wall, the complex K , and its polytopal realizationchange by a surgery which we describe below. Denote by P ol Old andby P ol New the polyhedra that realize K before and after the surgeryrespectfully.Let us look at the surgery in more details. Once a wall is crossed,some maximal by inclusion short set T ⊂ { , ..., n } turns to a long set,whereas its complement T = [ n +1] \ T becomes short. We conclude thatthe new complex K can be obtained from the old complex by removingsome of the cells and adding some new cells. The cells that get removedafter crossing the wall are labeled by ( ∗ , T, ∗ ). Here whereas the newcells that appear are labeled by ( ∗ , [ n ] \ T ). Here ” ∗ ” means just anyordered partition of the complement assuming that altogether we haveat least three parts.The cells that get removed form a subcomplex isomorphic to theboundary of the permutohedron Π n −| T | multiplied by a ( | T | − K converts this ball to the permutohedron Π | T | .So, we have the following Morse surgery: we cut out the cell subcomplex C = ( ∂ Π n −| T | ) × Π | T | , and patch instead the cell complex C = Π n −| T | × ∂ Π | T | along the identity mapping on their common boundary ∂ Π n −| T | × ∂ Π | T | .Denote by C := C ∪ C the union of these complexes. Combinatori-ally, we have C = ∂ (cid:16) Π n −| T | × Π | T | (cid:17) . C (taken with an appropriate orientation) relates the old and newpolyhedra. Namely, we have a homological sum: P ol New = P ol Old + C . This means that the new and old volumes are related by V ol ( P ol New ) = V ol ( P ol Old ) + V ol ( C ) . After geometrically realizing these complexes, we decompose the re-alization of Π n −| T | × Π | T | into the homological sum of bricks P i × P j ,where P i is an elementary brick from Π n −| T | , and P j is an elementarybrick from Π | T | . The first elementary brick P i corresponds to a tree on T , whereas P j corresponds to a tree on [ n + 1] \ T , or, equivalently, toa rooted forest on [ n ] \ T . In other words, each such pair ( P i , P j ) givesus a rooted forest F whose free tree is non-admissible.The brick P i × P j has a geometrical realization as the Minkowski sumof corresponding line segments. It contributes ( − n ) | M ( F ) | to V ol ( C ). Therefore, if the statement of the theorem is true for P ol Old , it isalso true for P ol New . (cid:3) Theorem 3. For a flexible ( n + 1) -polygon L , we have: V ol ( M ( L )) = √ n n X k =0 ( − k · a k · ( n − k ) n − , where a k is the number of ( k + 1) -element short subsets of [ n + 1] containing the entry ( n + 1) . Proof. Using Lemma 4, we first fix a number k and choose a long k -element subset of [ n ]. This can be done in a n − k ways. We put a treeon these vertices in k k − ways and arrive at V ol ( M ( L )) = 1 √ n n X k =1 a n − k · k k − X g is a rooted forest on ( n − k ) vertices ( − n ) C ( g ) · N ( F ) == 1 √ n n X k =1 a n − k · k k − X g is a rooted forest on ( n − k ) vertices ( − n ) C ( g ) =By the identity from Section 2.3 X g is a rooted forest on m vertices x C ( g ) = x · ( x + m ) m − , we get = 1 √ n n X k =1 a n − k ( − k ) k − · ( − n ) · k n − k − = −√ n n X k =1 a n − k k k − · ( − k ) n − k − == √ n n X k =1 a n − k k n − · ( − n − k . Interchanging k and n − k , we get the desired. (cid:3) Remark. Betty numbers β k = β k ( M ( L )) are expressed in terms of a k , see [2]: β k = a k + a n − k − . OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 17 Corollary 1. Assume n +1 = 2 m +1 . For the equilateral ( n +1) -linkage L = (1 , , ..., we have: V ol ( M ( L )) = √ m m X k =0 ( − k · (cid:18) mk (cid:19) · (2 m − k ) m − . Proof. Indeed, for the equilateral linkage, ”a short set” means ”a setwith cardinality ≤ m ”. Therefore a k = (cid:26) (cid:0) n − k (cid:1) , if k ≤ m ;0 , otherwise. (cid:3) Integer points counting for cyclopermutohedron Integer points counting for cyclopermutohedron: theoret-ical backgrounds. The first leading idea for integer points enumera-tion in a zonotope is to decompose it into elementary bricks, as we didin Section 1. However, unlike volume computation, we have to takeinto account the ”pieces” of all dimensions, including points. By thisreason we introduce semiopen bricks . The latter are Minkowski sumsof semiopen segments, see Figure 3. Figure 3. A semiopen segment and a semiopen rectan-gle. The dashed lines and white points are missing. Figure 4. The permutohedron Π splits into threesemiopen parallelograms, three semiopen segments, andone point. A zonotope decomposes in a disjoint union of semiopen bricks ofdimensions ranging from 0 to n . Example 2. Permutohedron Π n decomposes in a disjoint union ofsemiopen bricks that are in a one-to-one correspondence with forestson n labeled vertices. Each of the bricks contributes exactly one integerpoint, so for the number of integer points Λ , we have: Λ(Π n ) = number of forests on n labeled vertices. Below we almost literally repeat the arguments from Section 3. As-sume we have a convex zonotope Z ⊂ R n , that is, the Minkowski sumof linear segments { s i } mi =1 : Z = m M i =1 s i . For each subset I ⊂ [ m ] with | I | ≤ n , which gives linearly indepen-dent { s i } i ∈ I , denote by Z I the semiopen brick spanned by segments { s i } i ∈ I . It is well-known that Z can be partitioned into the union ofall such Z I , which implies immediatelyΛ( Z ) = X I ⊂ [ m ] ♯ ( Z I ) , where ♯ ( · ) denotes the number of integer points in a semiopen brickprovided that the brick is spanned by linearly independent vectors. Forlinearly dependent vectors we set ♯ := 0 . For positive integer numbers λ , ..., λ n let us sum up the dilatedsegments λ i s i . Clearly, we haveΛ (cid:16) m M i =1 λ i s i (cid:17) = X I ⊂ [ m ] ♯ ( Z I ) · Y i ∈ I λ i . For fixed s i , Λ is a polynomial in λ i , which counts not only thenumber of integer points in a convex zonotope (which originates frompositive λ i ), but also the number of integer points in a virtual zonotope,(which originates from any integer λ i , including negative ones), see[4, 8]. Remark. According to Khovanskii’s and Pukhlikov’s construction[4], given a lattice virtual polytope, each lattice point has a weight ,which is some (possibly, negative) integer number. The above definedΛ( · ) for virtual zonotopes counts the sum of weights. This fact general-izes the Erchart’s reciprocity law and has many other interpretations,such as Riemann-Roch Theorem for toric varieties. OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 19 We immediately have: Lemma 5. Let E = E n be the set of edges of the complete graph K n .For the cyclopermutohedron we have: Λ( CP n +1 ) = X ( I,M ): | I | + | M |≤ n − ( − | M | · ♯ (cid:0) M I q ij + M M r k (cid:1) Here I ranges over subsets of E , whereas M ranges over subsets of [ n ] . (cid:3) Our next aim is to give a formula for one single summand. Definition 4. A partial decorated forest F = ( G, M ) is a graph G = ([ n ] , I ) without cycles on n labeled vertices together with a set ofmarked vertices M ⊂ [ n ] such that the following conditions hold: (1) Number of marked vertices | M | + number of edges | I | is smalleror equal than n . (2) Each connected component of G has at most one marked vertex. We already know that decorated forests are in a bijection with lin-early independent ( n − { q ij , r k } (see Section 3). Therefore,partial decorated forests are in a bijection with linearly independentcollections of segments { q ij , r k } .From now on, we fix one particular partial decorated forest F andwork with the associated segments. Notation : F splits into a disjoint union of two forests: (1) a forest T = T ( F ) without marked vertices, which is called the free forest , and(2) a rooted forest R ( F ). In turn, T is a disjoint union of trees T j ( F ).As in the previous sections, C ( · ) denotes the number of connectedcomponents of a forest. In particular, C ( R ( F )) = | M | is the numberof marked vertices.In this notation we have: Lemma 6. For the number ♯ of integer points in the semiopen brickspanned by { q ij , r k } , we have: (1) If the segments in question do not come from a partial decoratedforest, then ♯ = 0 . (2) If the segments in question come from a decorated partial forest F with at least one marked vertex, then ♯ = n | M |− · gcd[ V ( T ) , . . . , V ( T C ( T ) )] , where | M | is the number marked vertices in F , T i are theconnected components of the free forest T , V ( T i ) is the numberof vertices in T i . (3) If the segments in question come from a decorated partial forest F with no marked vertices, then ♯ = 1 . For the proof of the lemma, see Section 6. (cid:3) Basing on the lemma, we obtain: Theorem 4. Define Φ( v ) = X T gcd[ { V ( T i ) } ] , where the sum ranges over all (non-rooted) forests T on v labeled ver-tices, T i are the trees in the forest T , and V ( · ) is the number of vertices.Then Λ( CP n +1 ) = ϕ ( n ) − n − X v =1 (cid:18) nv (cid:19) ( − v ) n − v − · Φ( v ) == Λ(Π n ) − n − X v =1 (cid:18) nv (cid:19) ( − v ) n − v − · Φ( v ) , where ϕ ( n ) is the number of (non-rooted) forests on n labeled vertices. Proof.(1) We count partial decorated forests with no marked points sep-arately. Altogether they contribute ϕ ( n ) = Λ(Π n ).(2) Next we choose v vertices of the free forest. This can be donein (cid:0) nv (cid:1) ways.(3) Each of the forests gives us its own gcd . Altogether they giveus Φ( v ).(4) Next, we count rooted forests on the remaining n − v vertices.Each forest is counted with multiplicity ( − n ) C ( F ) . The equality X f is a rooted forest on m vertices x C ( f ) = x · ( x + m ) m − (see Section 2.3) completes the proof. (cid:3) Examples: Λ( CP ) = 1,Λ( CP ) = 18. OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 21 Proof of Lemma 6. We fix a partial decorated forest and thecorresponding semiopen brick spanned by { q ij , r k } . The vectors r l willbe called long vectors , whereas q ij will be called short vectors .As the main tool, we shall use the following lemma, whose proofcomes from elementary linear algebra. Lemma 7. (1) The number of integer points ♯ ( { v i } ) doesn’t changeif we replace any v j by the vector v j + X i = j ( ± v i ) . (2) For an integer λ , we have: ♯ ( { λ · v , v , v ..., v k } ) = λ · ♯ ( { v , v , v ..., v k } ) . (3) Suppose there exists a coordinate x j such that among vectors { v i } only one vector (say, v ) has nonzero j th coordinate whichequals ± . This will be called the free coordinate. Then we canremove v from the collection of segments without changing thenumber of integer points: ♯ ( { v i } ) = ♯ ( { v i } i =1 ) . (4) Given a partial decorated forest, replace all the trees by pathtrees, keeping for each tree the set of its vertices. This manip-ulation does not change the value ♯ ( F ) . (5) Given one vector v = ( V , ..., V n ) , ♯ ( v ) = gcd[ { V i } ] , where gcd denotes the greatest common divisor. (cid:3) Reduction of a partial decorated forest (see Figure 5) goes as follows:Assume we have a partial decorated forest F .(1) Choose a marked vertex. We shall call it the principal markedvertex .(2) Join the principal marked vertex with each of the other markedvertices by an edge.(3) Remove all marks from the marked vertices that are not prin-cipal.(4) Replace the tree with the marked vertex by a path tree on thesame vertices. We arrive at a partial decorated forest F .Lemma 7 implies: Lemma 8. For a partial decorated forest F and its reduction F , wehave: ♯ ( F ) = n | M |− · ♯ ( F ) , where | M | = | M ( F ) | is the number of marked vertices in F . (cid:3) Figure 5. Reduction of a partial decorated forests.Grey balls denote the marked vertices.Now we are ready to calculate one single summand from Lemma5. We arrange the column vectors in a matrix: first come all the q ij ,after them come all the r k . The main idea is that the reduction processencodes the way of manipulating with the columns in the matrix. UsingLemma 7, (1), we can assume that all the P i are path trees.(1) Assume that the collection contains some long vector.The algorithm runs as follows: first, we take the long vectorwhich corresponds to the principal marked vertex and subtractit from all the other long vectors. Each of the long vectors(except for the first one) yields a multiple n and a new shortvector.Next, we subtract the short vectors from the (unique thatsurvived) long vector aiming at killing its coordinates. Finally,we get a matrix which allows to remove vectors using Lemma7, (3). Eventually we arrive at n | M |− · ♯ − V ( T )... − V ( T C ( T ) ) V ( T ) + · · · + V ( T C ( T ) ) = n | M |− · gcd[ V ( T ) , . . . , V ( T C ( T ) )] . (2) If there are no long vectors in the collection, we remove thevectors one by one using Lemma 7, (3), and arrive at ♯ = 1. (cid:3) OLUME AND LATTICE POINTS FOR CYCLOPERMUTOHEDRON 23 1 2 3 Figure 6. Partial decorated forests. Grey balls denotethe marked vertices. Examples. We exemplify below the reduction for three collectionsof vectors. Corresponding partial decorated forests are depicted in Fig.6. (1) Two free trees with V = 2 and V = 3, | M | = 1. ♯ − − − − − 10 0 1 − 10 0 − − 10 0 0 5 = ♯ − − − 30 0 1 − 10 0 − − 10 0 0 5 == 5 · ♯ − − − 30 0 1 00 0 − − 20 0 0 5 = ♯ − − = 1 . (2) One free tree with V = 4, | M | = 1. ♯ − − − − 10 1 − − 10 0 1 1 − 10 0 0 − − = ♯ − − − − 10 1 − − 10 0 1 1 − 20 0 0 − == ♯ − − − − 10 1 − − 30 0 1 1 00 0 0 − = ♯ − − − − 40 1 − − = = ♯ − − − 40 1 − − = ♯ − = 4 . (3) One free tree with V = 2, | M | = 1. ♯ − − − − − 10 1 − − 10 0 0 − − = ♯ − − − − 10 1 − − 20 0 0 − == ♯ − − − − 10 1 − = ♯ − − − = ♯ − = 2 . References [1] M. Farber, Invitation to topological robotics , European Mathematical Society,2008.[2] M. Farber and D. Sch¨utz, Homology of planar polygon spaces, Geom. Dedicata,125 (2007), 75-92.[3] A. Khovanskii, Newton polyhedra and toroidal varieties, Functional Analysisand Its Applications, 11(4):289–296, 1977.[4] A. Khovanskii and A. Pukhlikov, Finitely additive measures of virtual polytopes, St. Petersburg Math. J., Vol. 4, 2 (1993), 337-356.[5] G. Panina, Virtual polytopes and some classical problems , St. Petersburg Math.J., Vol. 14, 5 (2003), 823-834.[6] G. Panina, Moduli space of a planar polygonal linkage: a combinatorial descrip-tion , arXiv:1209.3241[7] G. Panina, Cyclopermutohedron , Proceedings of the Steklov Institute of Math-ematics, 2015, Vol. 288, 132-144.[8] G. 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