Integer points enumerator of hypergraphic polytopes

TL;DR

This paper introduces a quasisymmetric function invariant for hypergraphic polytopes that counts integer points and generalizes the Stanley chromatic symmetric function for graphs.

Contribution

It establishes a new combinatorial Hopf algebra framework linking hypergraphs to quasisymmetric functions, extending known graph invariants.

Findings

Defines a weighted quasisymmetric function for hypergraphic polytopes

Shows the invariant extends the Stanley chromatic symmetric function

Proves the universal morphism to quasisymmetric functions coincides with the enumerator

Abstract

For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function.

Equations

$$\mathbf{H}{|}_S=\{H\in \mathbf{H}:H\subseteq S\},$$

$$\mathbf{H}/S=\{H\setminus S:H\in \mathbf{H}\}.$$

$$[{\mathbf{H}}_1]\cdot [{\mathbf{H}}_2]=[{\mathbf{H}}_1\bigsqcup {\mathbf{H}}_2],$$

$$\Delta ([\mathbf{H}])=\sum _{S\subset [n]}[\mathbf{H}{|}_S]\otimes [\mathbf{H}/S].$$

$$S([\mathbb{H}])=\sum _{k\ge 1}(-1{)}^k\sum _{{\mathcal{L}}_k}\prod _{j=1}^k([\mathbb{H}]{\mid }_{I_j})/I_{j-1},$$

$$\Omega (a_1,a_2,\dots ,a_n)=\{(a_{\omega (1)},a_{\omega (2)},\dots ,a_{\omega (n)}):\omega \in {\mathfrak{S}}_n\},$$

$$M_{\mathcal{F}}=\sum _{\omega \in {\mathbb{Z}}_{+}^n\cap C_{\mathcal{F}}^{\circ }}{x}_{\omega },$$

$${\pi }_Q(\mathcal{F})=G\text{ if and only if }{C}_{\mathcal{F}}^{\circ }\subseteq C_G^{\circ },$$

$$F_q(Q)=\sum _{\omega \in {\mathbb{Z}}_{+}^n}{q}^{\dim ({\pi }_Q({\mathcal{F}}_{\omega }))}{x}_{\omega }=\sum _{\mathcal{F}\in L(P{e}^{n-1})}{q}^{\dim ({\pi }_Q(\mathcal{F}))}{M}_{\mathcal{F}},$$

$$f(Q,q)=(-1{)}^n\mathbf{ps}(F_{-q}(Q))(-1).$$

$$P_{\mathbf{H}}=\sum _{H\in \mathbf{H}}{\Delta }_H.$$

$${\mathrm{rk}}_{\mathbf{H}}(\mathcal{F})=\dim ({\pi }_{P_{\mathbf{H}}}(\mathcal{F})).$$

$$F_q(P_{\mathbf{H}})=\sum _{\mathcal{F}\in L(P{e}^{n-1})}{q}^{{\mathrm{rk}}_{\mathbf{H}}(\mathcal{F})}{M}_{\mathcal{F}}.$$

$${\zeta }_q([\mathbf{H}])=q^{\mathrm{rk}(\mathbf{H})}=q^{n-c(\mathbf{H})},$$

$${\Psi }_q([\mathbf{H}])=\sum _{\alpha \models n}({\zeta }_q{)}_{\alpha }([\mathbf{H}])M_{\alpha }.$$

$$\mathbf{H}/\mathcal{F}=\underset{i=1}{\overset{k}{⨆}}\mathbf{H}{|}_{F_i}/F_{i-1}.$$

$$({\zeta }_q{)}_{\alpha }([\mathbf{H}])=\sum _{\mathcal{F}:\mathrm{type}(\mathcal{F})=\alpha }\prod _{i=1}^k{q}^{\mathrm{rk}(\mathbf{H}{|}_{F_i}/F_{i-1})}=\sum _{\mathcal{F}:\mathrm{type}(\mathcal{F})=\alpha }{q}^{\mathrm{rk}(\mathbf{H}/\mathcal{F})},$$

$$\mathrm{rk}(\mathbf{H}/\mathcal{F})=\sum _{i=1}^k\mathrm{rk}(\mathbf{H}{|}_{F_i}/F_{i-1})=n-\sum _{i=1}^{k}c(\mathbf{H}{|}_{F_i}/F_{i-1}).$$

$${\Psi }_q([\mathbf{H}])=\sum _{\mathcal{F}\in L(P{e}^{n-1})}{q}^{\mathrm{rk}(\mathbf{H}/\mathcal{F})}{M}_{\mathrm{type}(\mathcal{F})}.$$

$$F_q(P_{\mathbf{H}})={\Psi }_q([\mathbf{H}]).$$

$${\mathrm{rk}}_{\mathbf{H}}(\mathcal{F})=\mathrm{rk}(\mathbf{H}/\mathcal{F}).$$

$$G=P_{\mathbf{H}/\mathcal{F}}.$$

$$f(P_{\mathbf{H}},q)=(-1{)}^n\mathbf{ps}\left({\Psi }_{-q}([\mathbf{H}])\right)(-1).$$

$$\sum _{i=1}^k\left(\genfrac{}{}{0ex}{}{n}{k-i,i,n-k}\right)q^{i-1}{M}_{(1)}^{k-i}\circ M_{(i)}\circ {\Psi }_q([{\mathbf{C}}_{n-k}]).$$

$$\sum _{0\le a<k<n-b\le n}\left(\genfrac{}{}{0ex}{}{n}{a,b,n-a-b}\right)q^{n-a-b-1}{M}_{(1)}^a\circ M_{(n-a-b)}\circ {\Psi }_q({\mathbf{C}}_b).$$

$$f(P_{{\mathbb{U}}_{n,k}},q)=\sum _{i=1}^k\left(\genfrac{}{}{0ex}{}{n}{k-i,i,n-k}\right)q^{i-1}f(P{e}^{n-k-1},q)+$$

$$+\sum _{0\le a<k<n-b\le n}\left(\genfrac{}{}{0ex}{}{n}{a,b,n-a-b}\right)q^{n-a-b-1}f(P{e}^{b-1},q).$$

$$F_q(P{S}^n)=F_q(P{S}^{n-1})M_{(1)}+(q-1)(F_q(P{S}^{n-1}){)}_{+1},$$

$$f_q(P{S}_n)=(2+q)f_q(P{S}_{n-1}),$$

$$P_{\Gamma }=\sum _{\{i,j\}\in \Gamma }{\Delta }_{e_i,e_j}.$$

Full text

Keywords: quasisymmetric function, hypergraph, hypergraphic polytope, combinatorial Hopf algebra

1 Introduction

The theory of combinatorial Hopf algebras developed by Aguiar, Bergeron and Sottile in the seminal paper [2] provides an algebraic framework for symmetric and quasisymmetric generating functions arising in enumerative combinatorics. Extensive studies of various combinatorial Hopf algebras are initiated recently [3],[4],[9],[10]. The geometric interpretation of the corresponding (quasi)symmetric functions was first given for matroids [4] and then for simple graphs [6] and building sets [7]. The quasisymmetric function invariants are expressed as integer points enumerators associated to generalized permutohedra. This class of polytopes introduced by Postnikov [11] is distinguished with rich combinatorial structure. The comprehensive treatment of weighted integer points enumerators associated to generalized permutohedra is carried out by Grujić et al. [8]. In this paper we consider a certain naturally defined non-cocommutative combinatorial Hopf algebra of hypergraphs and show that the derived quasisymmetric function invariant of hypergraphs is integer points enumerator of hypergraphic polytopes (Theorem 4.2).

2 Combinatorial Hopf algebra of hypergraphs $\mathcal{HG}$

A combinatorial Hopf algebra is a pair $(\mathcal{H},\zeta)$ of a graded connected Hopf algebra $\mathcal{H}=\oplus_{n\geq 0}\mathcal{H}_{n}$ over a field $\mathbb{k}$, whose homogeneous components $\mathcal{H}_{n},n\geq 0$ are finite-dimensional, and a multiplicative linear functional $\zeta:\mathcal{H}\rightarrow\mathbb{k}$ called character. We consider a combinatorial Hopf algebra structure on hypergraphs different from the chromatic Hopf algebra of hypergraphs studied in [10]. The difference is in the coalgebra structures based on different combinatorial constructions, which is manifested in (non)co-commutativity. It extends the Hopf algebra of building set studied by Grujić in [6]. This Hopf algebra of hypergraphs can be derived from the Hopf monoid structure on hypergraphs introduced in [1].

A hypergraph $\mathbf{H}$ on the vertex set $V$ is a collection of nonempty subsets $H\subseteq V$, called hyperedges. We assume that there are no ghost vertices, i.e. $\mathbf{H}$ contains all singletons $\{i\},i\in V$. A hypergraph $\mathbf{H}$ is connected if it can not be represented as a disjoint union of hypergraphs $\mathbf{H}_{1}\sqcup\mathbf{H}_{2}$. Every hypergraph $\mathbf{H}$ splits into its connected components. Let $c(\mathbb{H})$ be the number of connected components of $\mathbb{H}$. Hypergraphs $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ are isomorphic if there is a bijection of their sets of vertices that sends hyperedges to hyperedges. Let $\mathcal{HG}=\bigoplus_{n\geq 0}\mathcal{HG}_{n},$ where $\mathcal{HG}_{n}$ is the linear span of isomorphism classes $[\mathbf{H}]$ of hypergraphs on the set $[n]$.

Definition 2.1**.**

For a subset $S\subseteq[n]$ the restriction $\mathbb{H}|_{S}$ and the contraction $\mathbb{H}/S$ are defined by

[TABLE]

[TABLE]

Define a product and a coproduct on the linear space $\mathcal{HG}$ by

[TABLE]

[TABLE]

The straightforward checking shows that the space $\mathcal{HG}$ with the above operations and the unit $\eta:\mathbb{k}\rightarrow\mathcal{HG}$ given by $\eta(1)=[\mathbf{H}_{\emptyset}]$ (the empty hypergraph) and the counit $\epsilon:\mathcal{HG}\rightarrow\mathbb{k}$ which is the projection on the component $\mathcal{HG}_{0}=\mathbb{k}$, become a graded connected commutative and non-cocommutative bialgebra. Since graded connected bialgebras of finite type posses antipodes, $\mathcal{HG}$ is in fact a Hopf algebra. The formula for antipode $S:\mathcal{HG}\rightarrow\mathcal{HG}$ is derived from the general Takeuchi’s formula [12]

[TABLE]

where the inner sum goes over all chains of subsets $\mathcal{L}_{k}:\emptyset=I_{0}\subset I_{1}\subset\cdots\subset I_{k-1}\subset I_{k}=V$. Define a character $\zeta:\mathcal{HG}\rightarrow\mathbf{k}$ by $\zeta([\mathbb{H}])=1$ if $\mathbb{H}$ is discrete, i.e. contains only singletons and $\zeta([\mathbb{H}])=0$ otherwise. This determines the combinatorial Hopf algebra $(\mathcal{HG},\zeta)$.

3 Integer points enumerator

In this section we review the definition of the integer points enumerator of a generalized permutohedron introduced in [8].

For a point $(a_{1},a_{2},\ldots,a_{n})\in\mathbb{R}^{n}$ with increasing coordinates $a_{1}<\cdots<a_{n}$ let us define the set $\Omega(a_{1},a_{2},\ldots,a_{n})$ by

[TABLE]

where $\mathfrak{S}_{n}$ is the permutation group of the set $[n]$. The convex hull of the set $\Omega(a_{1},a_{2},\ldots,a_{n})$ is a standard $(n-1)-$dimensional permutohedron $Pe^{n-1}$. The $d-$dimensional faces of $Pe^{n-1}$ are in one-to-one correspondence with set compositions $\mathcal{C}=C_{1}|C_{2}|\cdots|C_{n-d}$ of the set $[n],$ see [11], Proposition 2.6. By this correspondence and the obvious correspondence between set compositions and flags of subsets we identify faces of $Pe^{n-1}$ with flags $\mathcal{F}:\emptyset=F_{0}\subset F_{1}\subset\cdots\subset F_{n-d}=[n]$. The dimension of a face and length of the corresponding flag is related by $\dim(\mathcal{F})=n-|\mathcal{F}|$.

The normal fan $\mathcal{N}(Pe^{n-1})$ of the standard permutohedron is the braid arrangement fan $\{x_{i}=x_{j}\}_{1\leq i<j\leq n}$ in the space $\mathbb{R}^{n}$. The dimension of the normal cone $C_{\mathcal{F}}$ at the face $\mathcal{F}$ is $\dim(C_{\mathcal{F}})\;=\;|\mathcal{F}|.$ The relative interior points $\omega\in C^{\circ}_{\mathcal{F}}$ are characterized by the condition that their coordinates are constant on $F_{i}\setminus F_{i-1}$ and increase $\omega|_{F_{i}\setminus F_{i-1}}<\omega|_{F_{i+1}\setminus F_{i}}$ with $i$. A positive integer vector $\omega\in\mathbb{Z}^{n}_{+}$ belongs to $C^{\circ}_{\mathcal{F}}$ if the weight function $\omega^{\ast}(x)\;=\;\langle\omega,x\rangle$ is maximized on $Pe^{n-1}$ along a face $\mathcal{F}$.

Definition 3.1**.**

For a flag $\mathcal{F}$ let $M_{\mathcal{F}}$ be the enumerator of interior positive integer points $\omega\in\mathbb{Z}^{n}_{+}$ of the corresponding cone $C_{\mathcal{F}}$

[TABLE]

where $\mathbb{x}_{\omega}=x_{\omega_{1}}x_{\omega_{2}}\cdots x_{\omega_{n}}$.

The enumerator $M_{\mathcal{F}}$ is a monomial quasisymmetric function depending only of the composition $\mathrm{type}(\mathcal{F})=(|F_{1}|,|F_{2}\setminus F_{1}|,\ldots,|F_{k}\setminus F_{k-1}|)$.

The fan $\mathcal{N}$ is a coaresement of $\mathcal{N}(Pe^{n-1})$ if every cone in $\mathcal{N}$ is a union of cones of $\mathcal{N}(Pe^{n-1})$. An $(n-1)-$dimensional generalized permutohedron $Q$ is a convex polytope whose normal fan $\mathcal{N}(Q)$ is a coaresement of $\mathcal{N}(Pe^{n-1})$. There is a map $\pi_{Q}:L(Pe^{n-1})\rightarrow L(Q)$ between face lattices given by

[TABLE]

where $C_{G}^{\circ}$ is the relative interior of the normal cone $C_{G}$ at the face $G\in L(Q).$

Definition 3.2**.**

For an $(n-1)-$generalized permutohedron $Q$ let $F_{q}(Q)$ be the weighted integer points enumerator

[TABLE]

where $\mathcal{F}_{\omega}$ is a unique face of $Pe^{n-1}$ containing $\omega$ in the relative interior.

Remark 3.3*.*

It is shown in [8, Theorem 4.4] that the enumerator $F_{q}(Q)$ contains the information about the $f$-vector of a generalized permutohedron $Q$. More precisely, the principal specialization of $F_{q}(Q)$ gives the $f$-polynomial of $Q$

[TABLE]

Recall that the principal specialization $\mathbf{ps}(F)(m)$ of a quasisymmetric function $F$ in variables $x_{1},x_{2},\ldots$ is a polynomial in $m$ obtained from the evaluation map at $x_{i}=1,i=1,\ldots,m$ and $x_{i}=0$ for $i>m$.

4 The hypergraphic polytope

For the standard basis vectors $e_{i},1\leq i\leq n$ in $\mathbb{R}^{n}$ let $\Delta_{H}=\mathrm{conv}\{e_{i}\,:\;i\in H\}$ be the simplex determined by a subset $H\subset[n]$. The hypergraphic polytope of a hypergraph $\mathbf{H}$ on $[n]$ is the Minkowski sum of simplices

[TABLE]

As generalized permutohedra can be described as the Minkowski sum of delated simplices (see [11]) we have that hypergraphic polytopes are generalized permutohedra. For the following description of $P_{\mathbf{H}}$ see [5, Section 1.5] and the references within it. Let $\mathbf{H}=\mathbf{H}_{1}\sqcup\mathbf{H}_{2}\sqcup\cdots\sqcup\mathbf{H}_{k}$ be the decomposition into connected components. Then $P_{\,\mathbf{H}}=P_{\,\mathbf{H}_{1}}\times P_{\,\mathbf{H}_{2}}\times\cdots\times P_{\,\mathbf{H}_{k}}$ and $\dim(P_{\,\mathbf{H}})=n-k.$ For connected hypergraphs $P_{\,\mathbf{H}}$ can be described as the intersection of the hyperplane $H_{\,\mathbf{H}}:=\left\{x\in\mathbb{R}^{n}\;\;:\;\;\sum_{i=1}^{n}x_{i}=|\mathbf{H}|\right\}$ with the halfspaces $H_{S,\,\geq}:=\left\{x\in\mathbb{R}^{n}\;\;:\;\;\sum_{i\in S}x_{i}\geq|\mathbf{H}|_{S}|\right\}$ corresponding to all proper subsets $S\subset[n]$. It follows that $P_{\,\mathbf{H}}$ can be obtained by iteratively cutting the standard simplex $\Delta_{[n]}$ by the hyperplanes $H_{S,\,\geq}$ corresponding to proper subsets $S$. For instance the standard permutohedron $Pe^{n-1}$ is a hypergraphic polytope $P_{\mathbf{C}_{n}}$ corresponding to the complete hypergraph $\mathbf{C}_{n}$ consisting of all subsets of $[n]$.

Definition 4.1**.**

For a connected hypergraph $\mathbf{H}$ the $\mathbf{H}-$rank is a map $\mathrm{rk}_{\,\mathbf{H}}:L(Pe^{n-1})\rightarrow\{0,1,\ldots,n-1\}$ given by

[TABLE]

Subsequently we deal only with connected hypergraphs. The quasisymmetric function $F_{q}(P_{\mathbf{H}})$ corresponding to a hypergraphic polytope $P_{\mathbf{H}}$, according to Definitions 3.2 and 4.1, depends only on the rank function

[TABLE]

We extend the ground field $\mathbb{k}$ to the field of rational function $\mathbb{k}(q)$ in a variable $q$ and consider the Hopf algebra $\mathcal{HG}$ over this extended field. Let $\mathrm{rk}(\mathbf{H})=n-c(\mathbf{H})$ for hypergraphs on $n$ vertices. Define a linear functional $\zeta_{q}:\mathcal{HG}\rightarrow\mathbf{k}(q)$ with

[TABLE]

which is obviously multiplicative. By the characterization of the combinatorial Hopf algebra of quasisymmetric functions $(QSym,\zeta_{Q})$ as a terminal object ([2, Theorem 4.1]) there exists a unique morphism of combinatorial Hopf alegbras $\Psi_{q}\;:\;(\mathcal{H},\;\zeta_{q})\;\rightarrow\;(\mathcal{Q}Sym,\;\zeta_{\mathcal{Q}})$ given on monomial basis by

[TABLE]

We determine the coefficients by monomial functions in the above expansion more explicitly. For a hypergraph $\mathbf{H}$ define its splitting hypergraph $\textbf{H}/\mathcal{F}$ by a flag $\mathcal{F}$ with

[TABLE]

The coefficient corresponding to a composition $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})\models n$ is a polynomial in $q$ determined by

[TABLE]

where the sum is over all flags $\mathcal{F}:\emptyset=:F_{0}\subset F_{1}\subset\cdots\subset F_{k}:=[n]$ of the type $\alpha$ and

[TABLE]

By this correspondence, we have

[TABLE]

Now we have two quasisymmetric functions associated to hypergraphs whose expansions in monomial bases are given by $(\ref{F_q})$ and $(\ref{Psi})$. We show that they actually coincide which describes the corresponding hypergraphic quasisymmetric invariant algebraically and geometrically.

Theorem 4.2**.**

For a connected hypergraph $\mathbf{H}$ the integer points enumerator $F_{q}(P_{\,\mathbf{H}})$ associated to a hypergraphic polytope and the quasisymmetric function $\Psi_{q}([\mathbf{H}])$ coincide

[TABLE]

Proof.

Let $\mathbf{H}$ be a connected hypergraph on the set $[n]$ and $\mathcal{F}:\emptyset=F_{0}\subset F_{1}\subset F_{2}\subset\cdots\subset F_{m}=[n]$ be a flag of subsets of $[n]$. It is sufficient to prove that

[TABLE]

For this we need to determine the face $G$ of the hypergraphic polytope $P_{\mathbf{H}}$ along which the weight function $\omega^{\ast}$ is maximized for an arbitrary $\omega\in C_{\mathcal{F}}^{\circ}$. Since $P_{\mathbf{H}}$ is the Minkowski sum of simplices $\Delta_{H}$ for $H\in\mathbf{H}$ the face $G$ is itself a Minkowski sum of the form $G=\sum_{H\in\mathbf{H}}(\Delta_{H})_{\mathcal{F}}$ where $(\Delta_{H})_{\mathcal{F}}$ is a unique face of $\Delta_{H}$ along which the weight function $\omega^{\ast}$ is maximized for $\omega\in C_{\mathcal{F}}^{\circ}$. Let $\omega=(\omega_{1},\ldots,\omega_{n})$ where $\omega_{i}=j$ if $i\in F_{j}\setminus F_{j-1}$ for $i=1,\ldots,n$. Then $\omega\in C_{\mathcal{F}}^{\circ}$ and we can convince that $(\Delta_{H})_{\mathcal{F}}=\Delta_{H\setminus F_{j-1}}$ where $j=\min\{k\mid H\subset F_{k}\}$. Denote by $\mathbf{H}_{j}$ the collection of all $H\in\mathbf{H}$ with $j=\min\{k\mid H\subset F_{k}\}$ for $j=1,\ldots,m$. We can represent the face $G$ as $G=\sum_{j=1}^{m}\sum_{H\in\mathbf{H}_{j}}\Delta_{H\setminus F_{j-1}},$ which shows that $G$ is precisely a hypergraphic polytope corresponding to the splitting hypergraph

[TABLE]

The equation $(\ref{jedan})$ follows from the fact that $\dim P_{\mathbf{H}/\mathcal{F}}=\mathrm{rk}(\mathbf{H}/\mathcal{F}),$ which is given by $(\ref{splitt})$.

∎

As a corollary, by Remark 3.3 and the equation $(\ref{face})$ within it, we can derive the $f$-polynomial of a hypergraphic polytope $P_{\mathbf{H}}$ in a purely algebraic way.

Corollary 4.3**.**

The $f-$polynomial of a hypergraphic polytope $P_{\,\mathbf{H}}$ is determined by the principal specialization

[TABLE]

We proceed with some examples and calculations.

Example 4.4**.**

Let $\mathbb{U}_{n,k}$ be the $k$-uniform hypergraph containing all $k$-elements subsets of $[n]$ with $k>1$. Divide flags into two families depending on whether they contain a $k$-elements subset. Let $\circ$ be a bilinear operation on quasisymmetric functions given on the monomial bases by concatenation $M_{\alpha}\circ M_{\beta}=M_{\alpha\cdot\beta}.$ The flags that contain $k$-elements subset contribute to $\Psi([\mathbb{U}_{n,k}])$ with

[TABLE]

The contribution to $\Psi([\mathbb{U}_{n,k}])$ of the remaining flags is

[TABLE]

By Corollary 4.3 since the principal specialization respects the operation $\circ$ it follows from $P_{\mathbf{C}_{m}}=Pe^{m-1}$ that

[TABLE]

[TABLE]

Example 4.5**.**

The hypergraphic polytope $PS^{n-1}$ corresponding to the hypergraph $\{[1],[2],\ldots,[n]\}$ is known as the Pitman-Stanley polytope. It is combinatorially equivalent to the $(n-1)$-cube [11, Proposition 8.10]. The following recursion is satisfied

[TABLE]

where +1 is given on monomial bases by $(M_{(i_{1},i_{2},\ldots,i_{k})})_{+1}=M_{(i_{1},i_{2}\ldots,\i_{k}+1)}$. It can be seen by dividing flags into two families according to the position of the element $n$. To a flag $\mathcal{F}:\emptyset=F_{0}\subset F_{1}\subset\cdots\subset F_{m}=[n]$ we associate the flag $\widetilde{\mathcal{F}}:\emptyset=F_{0}\subset F_{1}\setminus\{n\}\subset\cdots\subset F_{m}\setminus\{n\}=[n-1]$. If $n\in F_{k}$ for some $k<m$ then $\mathrm{rk}_{PS^{n}}(\mathcal{F})=\mathrm{rk}_{PS^{n-1}}(\mathcal{F})$ and if $n\notin F_{k}$ for $k<m$ then $\mathrm{rk}_{PS^{n}}(\mathcal{F})=\mathrm{rk}_{PS^{n-1}}(\mathcal{F})+1$. The principal specialization of the previous recursion formula gives

[TABLE]

consequently $f_{q}(PS_{n})=(2+q)^{n}$ which reflects the fact that $PS^{n}$ is an $n$-cube.

Example 4.6**.**

If $\Gamma$ is a simple graph, the corresponding hypergraphic polytope $P_{\Gamma}$ is the graphic zonotope

[TABLE]

Simple graphs generate the Hopf subalgebra of $\mathcal{HG}$ which is isomorphic to the chromatic Hopf algebra of graphs. Therefore $F_{q}(P_{\Gamma})$ is the $q$-analogue of the Stanley chromatic symmetric function of graphs introduced in [6].

Example 4.7**.**

Simplicial complexes generate another Hopf subalgebra of $\mathcal{HG}$ 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] that hypergraphic polytopes $P_{K}$ and $P_{K^{1}}$ corresponding to a simplicial complex $K$ and its $1$-skeleton $K^{1}$ are normally equivalent and therefore have the same enumerators

[TABLE]