aa r X i v : . [ m a t h . C O ] D ec Integer points enumerator ofhypergraphic polytopes
Marko Peˇsovi´c
Faculty of Civil Engineering, University of Belgrade [email protected]
Mathematics Subject Classifications: 05C65, 16T05, 52B11
Abstract
For a hypergraphic polytope there is a weighted quasisymmetricfunction which enumerates positive integer points in its normal fanand determines its f − polynomial. This quasisymmetric functioninvariant of hypergraphs extends the Stanley chromatic symmetricfunction of simple graphs. We consider a certain combinatorialHopf algebra of hypergraphs and show that universal morphism toquasisymmetric functions coincides with this enumerator function.We calculate the f − polynomial of uniform hypergraphic polytopes. Keywords : quasisymmetric function, hypergraph, hypergraphicpolytope, combinatorial Hopf algebra
The theory of combinatorial Hopf algebras developed by Aguiar, Bergeronand Sottile in the seminal paper [2] provides an algebraic framework forsymmetric and quasisymmetric generating functions arising in enumerativecombinatorics. Extensive studies of various combinatorial Hopf algebras areinitiated recently [3],[4],[9],[10]. The geometric interpretation of the corre-sponding (quasi)symmetric functions was first given for matroids [4] and thenfor simple graphs [6] and building sets [7]. The quasisymmetric function in-variants are expressed as integer points enumerators associated to generalized1ermutohedra. This class of polytopes introduced by Postnikov [11] is distin-guished with rich combinatorial structure. The comprehensive treatment ofweighted integer points enumerators associated to generalized permutohedrais carried out by Gruji´c et al. [8]. In this paper we consider a certain natu-rally defined non-cocommutative combinatorial Hopf algebra of hypergraphsand show that the derived quasisymmetric function invariant of hypergraphsis integer points enumerator of hypergraphic polytopes (Theorem 4.2). HG A combinatorial Hopf algebra is a pair ( H , ζ ) of a graded connected Hopf alge-bra H = ⊕ n ≥ H n over a field k , whose homogeneous components H n , n ≥ ζ : H → k called character . We consider a combinatorial Hopf algebra structure on hy-pergraphs different from the chromatic Hopf algebra of hypergraphs studiedin [10]. The difference is in the coalgebra structures based on different com-binatorial constructions, which is manifested in (non)co-commutativity. Itextends the Hopf algebra of building set studied by Gruji´c in [6]. This Hopfalgebra of hypergraphs can be derived from the Hopf monoid structure onhypergraphs introduced in [1].A hypergraph H on the vertex set V is a collection of nonempty subsets H ⊆ V , called hyperedges . We assume that there are no ghost vertices, i.e. H contains all singletons { i } , i ∈ V . A hypergraph H is connected if it can notbe represented as a disjoint union of hypergraphs H ⊔ H . Every hypergraph H splits into its connected components. Let c ( H ) be the number of connectedcomponents of H . Hypergraphs H and H are isomorphic if there is abijection of their sets of vertices that sends hyperedges to hyperedges. Let HG = L n ≥ HG n , where HG n is the linear span of isomorphism classes [ H ]of hypergraphs on the set [ n ]. Definition 2.1.
For a subset S ⊆ [ n ] the restriction H | S and the contraction H /S are defined by H | S = { H ∈ H : H ⊆ S } , H /S = { H \ S : H ∈ H } . product and a coproduct on the linear space HG by[ H ] · [ H ] = [ H ⊔ H ] , ∆([ H ]) = X S ⊂ [ n ] [ H | S ] ⊗ [ H /S ] . The straightforward checking shows that the space HG with the above op-erations and the unit η : k → HG given by η (1) = [ H ∅ ] (the empty hyper-graph) and the counit ǫ : HG → k which is the projection on the component HG = k , become a graded connected commutative and non-cocommutativebialgebra. Since graded connected bialgebras of finite type posses antipodes, HG is in fact a Hopf algebra. The formula for antipode S : HG → HG isderived from the general Takeuchi’s formula [12] S ([ H ]) = X k ≥ ( − k X L k k Y j =1 ([ H ] | I j ) /I j − , where the inner sum goes over all chains of subsets L k : ∅ = I ⊂ I ⊂ · · · ⊂ I k − ⊂ I k = V . Define a character ζ : HG → k by ζ ([ H ]) = 1 if H is discrete,i.e. contains only singletons and ζ ([ H ]) = 0 otherwise. This determines thecombinatorial Hopf algebra ( HG , ζ ). In this section we review the definition of the integer points enumerator of ageneralized permutohedron introduced in [8].For a point ( a , a , . . . , a n ) ∈ R n with increasing coordinates a < · · · < a n let us define the set Ω( a , a , . . . , a n ) byΩ( a , a , . . . , a n ) = { ( a ω (1) , a ω (2) , . . . , a ω ( n ) ) : ω ∈ S n } , where S n is the permutation group of the set [ n ]. The convex hull of the setΩ( a , a , . . . , a n ) is a standard ( n − − dimensional permutohedron P e n − .The d − dimensional faces of P e n − are in one-to-one correspondence withset compositions C = C | C | · · · | C n − d of the set [ n ] , see [11], Proposition2.6. By this correspondence and the obvious correspondence between setcompositions and flags of subsets we identify faces of P e n − with flags F :3 = F ⊂ F ⊂ · · · ⊂ F n − d = [ n ]. The dimension of a face and length of thecorresponding flag is related by dim( F ) = n − |F | .The normal fan N ( P e n − ) of the standard permutohedron is the braidarrangement fan { x i = x j } ≤ i For a flag F let M F be the enumerator of interior positiveinteger points ω ∈ Z n + of the corresponding cone C F M F = X ω ∈ Z n + ∩ C ◦F x ω , where x ω = x ω x ω · · · x ω n .The enumerator M F is a monomial quasisymmetric function depending onlyof the composition type( F ) = ( | F | , | F \ F | , . . . , | F k \ F k − | ).The fan N is a coaresement of N ( P e n − ) if every cone in N is a union ofcones of N ( P e n − ). An ( n − − dimensional generalized permutohedron Q is a convex polytope whose normal fan N ( Q ) is a coaresement of N ( P e n − ).There is a map π Q : L ( P e n − ) → L ( Q ) between face lattices given by π Q ( F ) = G if and only if C ◦F ⊆ C ◦ G , where C ◦ G is the relative interior of the normal cone C G at the face G ∈ L ( Q ) . Definition 3.2. For an ( n − − generalized permutohedron Q let F q ( Q ) bethe weighted integer points enumerator F q ( Q ) = X ω ∈ Z n + q dim( π Q ( F ω )) x ω = X F∈ L ( P e n − ) q dim( π Q ( F )) M F , where F ω is a unique face of P e n − containing ω in the relative interior. Remark . It is shown in [8, Theorem 4.4] that the enumerator F q ( Q ) con-tains the information about the f -vector of a generalized permutohedron Q .More precisely, the principal specialization of F q ( Q ) gives the f -polynomialof Q ( Q, q ) = ( − n ps ( F − q ( Q ))( − . (1)Recall that the principal specialization ps ( F )( m ) of a quasisymmetric func-tion F in variables x , x , . . . is a polynomial in m obtained from the evalu-ation map at x i = 1 , i = 1 , . . . , m and x i = 0 for i > m . For the standard basis vectors e i , ≤ i ≤ n in R n let ∆ H = conv { e i : i ∈ H } be the simplex determined by a subset H ⊂ [ n ]. The hypergraphic polytope of a hypergraph H on [ n ] is the Minkowski sum of simplices P H = X H ∈ H ∆ H . As generalized permutohedra can be described as the Minkowski sum of de-lated simplices (see [11]) we have that hypergraphic polytopes are generalizedpermutohedra. For the following description of P H see [5, Section 1.5] andthe references within it. Let H = H ⊔ H ⊔ · · · ⊔ H k be the decompo-sition into connected components. Then P H = P H × P H × · · · × P H k and dim( P H ) = n − k. For connected hypergraphs P H can be described asthe intersection of the hyperplane H H := { x ∈ R n : P ni =1 x i = | H |} withthe halfspaces H S, ≥ := (cid:8) x ∈ R n : P i ∈ S x i ≥ | H | S | (cid:9) corresponding to allproper subsets S ⊂ [ n ]. It follows that P H can be obtained by iterativelycutting the standard simplex ∆ [ n ] by the hyperplanes H S, ≥ correspondingto proper subsets S . For instance the standard permutohedron P e n − is ahypergraphic polytope P C n corresponding to the complete hypergraph C n consisting of all subsets of [ n ]. Definition 4.1. For a connected hypergraph H the H − rank is a map rk H : L ( P e n − ) → { , , . . . , n − } given byrk H ( F ) = dim( π P H ( F )) . Subsequently we deal only with connected hypergraphs. The quasisymmetricfunction F q ( P H ) corresponding to a hypergraphic polytope P H , according toDefinitions 3.2 and 4.1, depends only on the rank function5 q ( P H ) = X F∈ L ( P e n − ) q rk H ( F ) M F . (2)We extend the ground field k to the field of rational function k ( q ) in avariable q and consider the Hopf algebra HG over this extended field. Letrk( H ) = n − c ( H ) for hypergraphs on n vertices. Define a linear functional ζ q : HG → k ( q ) with ζ q ([ H ]) = q rk( H ) = q n − c ( H ) , which is obviously multiplicative. By the characterization of the combina-torial Hopf algebra of quasisymmetric functions ( QSym, ζ Q ) as a terminalobject ([2, Theorem 4.1]) there exists a unique morphism of combinatorialHopf alegbras Ψ q : ( H , ζ q ) → ( Q Sym, ζ Q ) given on monomial basis byΨ q ([ H ]) = X α | = n ( ζ q ) α ([ H ]) M α . We determine the coefficients by monomial functions in the above expansionmore explicitly. For a hypergraph H define its splitting hypergraph H / F bya flag F with H / F = k G i =1 H | F i /F i − . The coefficient corresponding to a composition α = ( α , α , . . . , α k ) | = n is apolynomial in q determined by( ζ q ) α ([ H ]) = X F : type( F )= α k Y i =1 q rk( H | Fi /F i − ) = X F : type( F )= α q rk( H / F ) , where the sum is over all flags F : ∅ =: F ⊂ F ⊂ · · · ⊂ F k := [ n ] of thetype α andrk( H / F ) = k X i =1 rk( H | F i /F i − ) = n − k X i =1 c ( H | F i /F i − ) . (3)By this correspondence, we haveΨ q ([ H ]) = X F∈ L ( P e n − ) q rk( H / F ) M type( F ) . (4)6ow we have two quasisymmetric functions associated to hypergraphswhose expansions in monomial bases are given by (2) and (4). We showthat they actually coincide which describes the corresponding hypergraphicquasisymmetric invariant algebraically and geometrically. Theorem 4.2. For a connected hypergraph H the integer points enumer-ator F q ( P H ) associated to a hypergraphic polytope and the quasisymmetricfunction Ψ q ([ H ]) coincide F q ( P H ) = Ψ q ([ H ]) . Proof. Let H be a connected hypergraph on the set [ n ] and F : ∅ = F ⊂ F ⊂ F ⊂ · · · ⊂ F m = [ n ] be a flag of subsets of [ n ]. It is sufficient to provethat rk H ( F ) = rk( H / F ) . (5)For this we need to determine the face G of the hypergraphic polytope P H along which the weight function ω ∗ is maximized for an arbitrary ω ∈ C ◦F .Since P H is the Minkowski sum of simplices ∆ H for H ∈ H the face G is itselfa Minkowski sum of the form G = P H ∈ H (∆ H ) F where (∆ H ) F is a uniqueface of ∆ H along which the weight function ω ∗ is maximized for ω ∈ C ◦F . Let ω = ( ω , . . . , ω n ) where ω i = j if i ∈ F j \ F j − for i = 1 , . . . , n . Then ω ∈ C ◦F and we can convince that (∆ H ) F = ∆ H \ F j − where j = min { k | H ⊂ F k } .Denote by H j the collection of all H ∈ H with j = min { k | H ⊂ F k } for j = 1 , . . . , m . We can represent the face G as G = P mj =1 P H ∈ H j ∆ H \ F j − , which shows that G is precisely a hypergraphic polytope corresponding tothe splitting hypergraph G = P H / F . The equation (5) follows from the fact that dim P H / F = rk( H / F ) , which isgiven by (3).As a corollary, by Remark 3.3 and the equation (1) within it, we can derivethe f -polynomial of a hypergraphic polytope P H in a purely algebraic way. Corollary 4.3. The f − polynomial of a hypergraphic polytope P H is deter-mined by the principal specialization f ( P H , q ) = ( − n ps (Ψ − q ([ H ])) ( − . 7e proceed with some examples and calculations. Example 4.4. Let U n,k be the k -uniform hypergraph containing all k -elements subsets of [ n ] with k > 1. Divide flags into two families dependingon whether they contain a k -elements subset. Let ◦ be a bilinear operationon quasisymmetric functions given on the monomial bases by concatenation M α ◦ M β = M α · β . The flags that contain k -elements subset contribute toΨ([ U n,k ]) with k X i =1 (cid:18) nk − i, i, n − k (cid:19) q i − M k − i (1) ◦ M ( i ) ◦ Ψ q ([ C n − k ]) . The contribution to Ψ([ U n,k ]) of the remaining flags is X ≤ a The hypergraphic polytope P S n − corresponding to the hy-pergraph { [1] , [2] , . . . , [ n ] } is known as the Pitman-Stanley polytope. It iscombinatorially equivalent to the ( n − F q ( P S n ) = F q ( P S n − ) M (1) + ( q − F q ( P S n − )) +1 , where +1 is given on monomial bases by ( M ( i ,i ,...,i k ) ) +1 = M ( i ,i ..., ı k +1) . Itcan be seen by dividing flags into two families according to the position ofthe element n . To a flag F : ∅ = F ⊂ F ⊂ · · · ⊂ F m = [ n ] we associatethe flag e F : ∅ = F ⊂ F \ { n } ⊂ · · · ⊂ F m \ { n } = [ n − n ∈ F k forsome k < m then rk P S n ( F ) = rk P S n − ( F ) and if n / ∈ F k for k < m then8k P S n ( F ) = rk P S n − ( F ) + 1. The principal specialization of the previousrecursion formula gives f q ( P S n ) = (2 + q ) f q ( P S n − ) , consequently f q ( P S n ) = (2+ q ) n which reflects the fact that P S n is an n -cube. Example 4.6. If Γ is a simple graph, the corresponding hypergraphic poly-tope P Γ is the graphic zonotope P Γ = X { i,j }∈ Γ ∆ e i ,e j . Simple graphs generate the Hopf subalgebra of HG which is isomorphic tothe chromatic Hopf algebra of graphs. Therefore F q ( P Γ ) is the q -analogue ofthe Stanley chromatic symmetric function of graphs introduced in [6]. Example 4.7. Simplicial complexes generate another Hopf subalgebra of HG which is isomorphic to the Hopf algebra of simplicial complexes introduced in[10] and studied more extensively in [3]. It is shown in [1, Lemma 21.2] thathypergraphic polytopes P K and P K corresponding to a simplicial complex K and its 1-skeleton K are normally equivalent and therefore have the sameenumerators F q ( P K ) = F q ( P K ) . References [1] M. 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