# Real toric manifolds associated with chordal nestohedra

Suyoung Choi  Department of mathematics, Ajou University, 206, World cup-ro,
Yeongtong-gu, Suwon 16499, Republic of Korea<EMAIL_ADDRESS>and  Younghan
Yoon  Department of mathematics, Ajou University, 206, World cup-ro,
Yeongtong-gu, Suwon 16499, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

This paper investigates the rational Betti numbers of real toric manifolds
associated with chordal nestohedra. We consider the poset topology of a
specific poset induced from a chordal building set, and show its EL-
shellability. Based on this, we present an explicit description using
alternating $\mathcal{B}$-permutations for a chordal building set
$\mathcal{B}$, transforming the computing Betti numbers into a counting
problem. This approach allows us to compute the $a$-number of a finite simple
graph through permutation counting when the graph is chordal. In addition, we
provide detailed computations for specific cases such as real Hochschild
varieties corresponding to Hochschild polytopes.

###### Key words and phrases:

homology group, toric topology, real toric varieties, chordal nestohedra,
poset topology, EL-shellability, $\mathcal{B}$-permutations, alternating
permutations, chordal graphs, graph associahedra, Stanley-Pitman polytopes,
Hochschild polytopes

###### 2020 Mathematics Subject Classification:

57S12, 14M25, 57N65, 05A05, 52B22

This work was supported by the National Research Foundation of Korea Grant
funded by the Korean Government (NRF-2021R1A6A1A10044950).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Real toric manifolds associated with nestohedra
    2. 2.2 Graph associahedra
    3. 2.3 Hochschild polytopes
  3. 3 EL-shellability of the nested set complex
  4. 4 Alternating $\mathcal{B}$-permutations
  5. 5 Proof of the main theorem
  6. 6 Examples
    1. 6.1 Permutohedra
    2. 6.2 Associahedra
    3. 6.3 Stellohedra
    4. 6.4 Hochschild polytopes
    5. 6.5 Cyclohedra
  7. 7 Open problems on the nested set complex
    1. 7.1 Venustus building sets
    2. 7.2 Unimodality and multiplicative structures

##  1\. Introduction

A _toric variety_ $X$ is a normal complex algebraic variety containing the
algebraic torus $(\mathbb{C}\setminus\\{O\\})^{n}$ on itself extends to an
action on $X$. A smooth compact toric variety $X$ is called a _toric manifold_
, and the real locus $X^{\mathbb{R}}$ of $X$, that is, the fixed point set of
$X$ by the canonical involution induced from a complex conjugation, is called
a _real toric manifold_. By the fundamental theorem of toric geometry, there
is a one-to-one correspondence between the class of $n$-dimensional toric
manifolds and the class of non-singular complete fans in $\mathbb{R}^{n}$. It
is well known [16] that a toric variety is projective if and only if its
associated fan is the normal fan of some simple polytope. An $n$-dimensional
simple polytope is said to be _Delzant_ if for each vertex $v$ the outward
normal vectors of the facets containing $v$ can be chosen so that they form an
integral basis for $\mathbb{Z}^{n}$. Since the normal fan of a Delzant
polytope $P$ is a non-singular complete fan, $P$ induces the projective toric
manifold and the corresponding real toric manifold.

Our investigation centers on some family of Delzant polytopes known as
_nestohedra_ , which has been well studied in various literature, such as
[35], [36], [11], [33], [23], [28], [24], [20], and [8]. For a finite set $S$
of natural numbers, a _building set_ on $S$ is a collection of nonempty
subsets of $S$ which includes every singleton of $S$ and satisfies the
following condition:

$I,J\in\mathcal{B}\text{ and }I\cap J\neq\emptyset\Rightarrow I\cup
J\in\mathcal{B}.$

The notion of building sets is defined in [18] with respect to a hyperplane
arrangement of vector spaces. An inclusion-maximal subset in $\mathcal{B}$ is
called a _connected component_ of $\mathcal{B}$. If there is a unique
connected component of $\mathcal{B}$, a building set $\mathcal{B}$ is said to
be _connected_.

A _nestohedron_ $P_{\mathcal{B}}$ associated with $\mathcal{B}$ is defined as
the Minkowski sum of simplices

$P_{\mathcal{B}}=\sum_{I\in\mathcal{B}}{\operatorname{conv}(I)},$

where $\operatorname{conv}(I)$ is the convex hull of the $i$th coordinate
vectors for all $i\in I$.

Since $P_{\mathcal{B}}$ is Delzant as shown in [35], it induces a projective
toric manifold. We refer to this toric manifold as the _toric manifold
associated with a building set $\mathcal{B}$_ and denoted it by
$X_{\mathcal{B}}$, and its corresponding real toric manifold by
$X^{\mathbb{R}}_{\mathcal{B}}$. For a connected building set $\mathcal{B}$ on
$S$ of cardinality $n+1$, the associated toric manifold $X_{\mathcal{B}}$ is
of dimension $n$. By [36, Remark 6.7], if
$\mathcal{B}_{1},\mathcal{B}_{2},\ldots,\mathcal{B}_{\ell}$ are the connected
components of a building set $\mathcal{B}$, then $P_{\mathcal{B}}$ is
isomorphic to the direct product

$P_{\mathcal{B}_{1}}\times\cdots\times P_{\mathcal{B}_{\ell}}.$

Moreover, by [16, Theorem 2.4.7], the associated toric manifold
$X_{\mathcal{B}}$ is isomorphic to the product
$X_{\mathcal{B}_{1}}\times\cdots\times X_{\mathcal{B}_{\ell}}$ as projective
toric manifolds. Hence, to study toric manifolds or real toric manifolds
associated with building sets, it is sufficient to consider the case when the
building set is connected.

As established in [17], for a building set $\mathcal{B}$, the rational Betti
numbers of $X_{\mathcal{B}}$ and the $\mathbb{Z}_{2}$-Betti numbers of
$X^{\mathbb{R}}_{\mathcal{B}}$ are completely determined from the numbers of
faces of $P_{\mathcal{B}}$, where $\mathbb{Z}_{2}$ is the field of order $2$.
Particularly, in the case where $\mathcal{B}$ is a _chordal building set_ on
$S$, that is,

$I=\\{i_{1}<\cdots<i_{r}\\}\in\mathcal{B}\Rightarrow\\{i_{s},\ldots,i_{r}\\}\in\mathcal{B}\text{
for each $1<s<r$},$

there is an explicit description, established in [36], of the face numbers of
$P_{\mathcal{B}}$ in terms of permutations on $S$. Note that this description
confirms that the Gal’s conjecture [22] is true for chordal nestohedra. To
discuss more details, we briefly introduce some notions for a building set
$\mathcal{B}$ on $S$:

  * •

For any subset $I$ of $S$, $\mathcal{B}|_{I}=\\{J\in\mathcal{B}\colon J\subset
I\\}$ is a _restricted building set_ of $\mathcal{B}$ to $I$.

  * •

A (one-line notation) permutation $x=(x_{1}x_{2}\cdots x_{k})$ on $S$ is a
_$\mathcal{B}$ -permutation_ if, for each $1\leq i\leq k$, there is
$J\in\mathcal{B}|_{\\{x_{1},x_{2},\ldots,x_{i}\\}}$ such that $J$ contains
both $x_{i}$ and $\max{\\{x_{1},x_{2},\ldots,x_{i}\\}}$.

From now on in this paper, for a topological space $M$, $\beta_{k}(M)$ will
simply be referred to as the $k$th rational Betti number of $M$. From [17] and
[36], for a connected chordal building set $\mathcal{B}$ on
$[n+1]=\\{1,2,\ldots,n+1\\}$, the $k$th Betti numbers of $X_{\mathcal{B}}$ is

(1.1)
$\beta_{k}(X_{\mathcal{B}})=\begin{cases}\\#\text{$\mathcal{B}$-permutations
with $\frac{k}{2}$ descents},&\mbox{if $0\leq k\leq 2n$ is even,}\\\
0,&\mbox{otherwise},\end{cases}$

where a _descent_ of a permutation $(x_{1}x_{2}\cdots x_{n})$ on
$[n+1]=\\{1,\ldots,n\\}$ is $1\leq i<n$ where $x_{i}>x_{i+1}$.

It is also natural to consider Betti number formulas for real toric manifold
$X^{\mathbb{R}}_{\mathcal{B}}$ associated with $\mathcal{B}$. Although the
$\mathbb{Z}_{2}$-Betti numbers of $X_{\mathcal{B}}^{\mathbb{R}}$ can also be
computed in the same manner as in (1.1), the computation of its rational Betti
numbers is significantly more complicated in general.

One typical and important example of a chordal building set is the maximal
building set $\mathcal{B}$ on $[n+1]$, that is, $\mathcal{B}$ is the set of
nonempty subsets of $[n+1]$. The corresponding polytope is known as a
_permutohedron_ and has dimension $n$. Likewise, the associated toric variety,
having a complex dimension of $n$, is referred to as the _Coxeter variety_ of
type $A_{n}$, denoted by $X_{A_{n}}$. In many literature, it is also known as
a _permutohedral variety_. Furthermore, its real counterpart
$X_{\mathcal{B}}^{\mathbb{R}}$ of dimension $n$ is called a _real Coxeter
variety_ $X_{A_{n}}^{\mathbb{R}}$ of type $A_{n}$ or a _real permutohedral
variety_. In his paper [25], Henderson employed a geometric approach to
compute the $k$th Betti numbers of $X_{A_{n}}^{\mathbb{R}}$, which can be
expressed as follows:

$\beta_{k}(X^{\mathbb{R}}_{A_{n}})=\binom{n+1}{2k}a_{2k},$

where $a_{2k}$ denotes the $2k$th Euler zigzag number (A000111 in [37]),
signifying the number of alternating permutations of length $2k$. Later, this
formula is revisited by several studies including [42], [11], [7], which
explored poset topologies based on the formula established in [43] and [12].
Additionally, the multiplicative structure of the cohomology ring of
$X_{A_{n}}^{\mathbb{R}}$ has been completely described in terms of alternating
permutations in our previous work [13].

One observes that a $\mathcal{B}$-permutation for the maximal building set
$\mathcal{B}$ on $[n+1]$ is an ordinary permutation. In other words, in this
case, the Betti number of
$X_{\mathcal{B}}^{\mathbb{R}}=X_{A_{n}}^{\mathbb{R}}$ can be described using
alternating ($\mathcal{B}$-)permutations, whereas that of
$X_{\mathcal{B}}=X_{A_{n}}$ is describable using permutations as in (1.1).
This observation motivates us to investigate whether the Betti numbers of
$X_{\mathcal{B}}^{\mathbb{R}}$ for a general chordal building set
$\mathcal{B}$ can similarly be described by alternating
$\mathcal{B}$-permutations, where an _alternating $\mathcal{B}$-permutation_
is a $\mathcal{B}$-permutation $x=(x_{1}x_{2}\cdots x_{n})$ such that
$x_{1}>x_{2}<x_{3}>\cdots$.

Now, we introduce the main result of this paper, and the proof will be given
in Section 5.

######  Theorem 1.1.

Let $\mathcal{B}$ be a connected chordal building set on
$[n+1]=\\{1,2,\ldots,n+1\\}$. The $k$th Betti number of
$X^{\mathbb{R}}_{\mathcal{B}}$ is

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}})=\sum_{I\in{[n+1]\choose
2k}}{\\#\text{alternating $\mathcal{B}|_{I}$-permutations}}.$

It should be noted that the condition of the chordality of a building set is
necessary. An example introduced in Section 6.5 demonstrates that Theorem 1.1
does not hold for an arbitrary building set even for small dimensions $n$.

To prove the main theorem, for a chordal building set $\mathcal{B}$ on
$[n+1]$, we need to consider the specific poset
$\hat{\mathcal{P}}_{\mathcal{B}}$ consisting of all subsets $I$ of $[n+1]$
such that $\mathcal{B}|_{I}$ does not admit odd components. The key part is
showing that the poset $\hat{\mathcal{P}}_{\mathcal{B}}$ is EL-shellable in
Section 3. EL-shellability has been extensively studied in the literature,
such as in [29], [4], [26], and [27]. The authors wish to emphasize that this
provides a non-trivial example of naturally occurring EL-shellable complexes.
Due to EL-shellability of $\hat{\mathcal{P}}_{\mathcal{B}}$, in Section 4, we
deal with the poset topology of $\hat{\mathcal{P}}_{\mathcal{B}}$ in terms of
alternating $\mathcal{B}$-permutations. The full proof of Theorem 1.1 will be
given in Section 5.

In Section 6, we apply the main result in several remarkable chordal
nestohedra, such as _permutohedra_ , _associahedra_ (or _Stasheff polytopes_),
_stellohedra_ , _Stanley-Pitman polytopes_ [41], and _Hochschild polytopes_
[34]. See Figure 1 for an illustrated diagram.

NestohedrdaChordal nestohedrdaGraph associahedraHochschild PolytopesStanley-
PitmanPolytopesPermutohedraAssoicahedraStellohedraCyclohedra Figure 1.
Categories of nestohedra

The first three are contained the class of _graph associahedra_ [5], that is,
nestohedra whose building set $\mathcal{B}=\mathcal{B}(G)$ consists of the
connected induced subgraph of a finite simple graph $G$. For a finite simple
graph $G$, there is a graph invariant $a(G)$, known as _the $a$-number_ of
$G$, as introduced in [11]. The Betti numbers of
$X^{\mathbb{R}}_{\mathcal{B}(G)}$ is determined by the $a$-numbers of
subgraphs of $G$. See Section 2.2 for more details.

A graph is said to be _chordal_ if every cycle containing four or more
vertices has a chord, which is an edge that connects two nonadjacent vertices
of the cycle. According to [36], a finite simple graph $G$ is a _chordal
graph_ with vertices labeled _perfect elimination ordering_ (refer Section
2.2) if and only if its building set $\mathcal{B}(G)$ is chordal. For a
chordal graph $G$ with a perfect elimination ordering, Theorem 1.1 provides a
combinatorial interpretation of the $a$-number $a(G)$ via
$\mathcal{B}(G)$-permutations. It is indeed beneficial because the computation
of $a(G)$ was transformed into a counting problem, whereas the original
definition of $a(G)$ is described recursively. In particular, the results of
Corollary 1.2 are equal to the $a$-numbers of _complete graphs_ , _path
graphs_ , and _star graphs_ , which correspond permutohedra, associahedra, and
stellohedra, respectively.

######  Corollary 1.2.

  1. (1)

For a complete graph $K_{n+1}$ having $n+1$ vertices,

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}(K_{n+1})})=\sum_{I\in{[n+1]\choose
2k}}\\#\text{alternating permutations on $I$}.$

  2. (2)

For a path graph $G=P_{n+1}$ having $n+1$ vertices labeled consecutively as
$1,\ldots,n+1$,

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}(P_{n+1})})=\sum_{I\in{[n+1]\choose
2k}}\prod_{1\leq i\leq r_{I}}\\#\text{$312$-avoiding alternating permutations
on $I_{i}$},$

where the induced subgraph $G|_{I}$ consists of the even order components
$G|_{I_{1}},\ldots,G|_{I_{r_{I}}}$.

  3. (3)

For a star graph $K_{1,n}$ having $n+1$ vertices with the central vertex
labeled as $n+1$,

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}(P_{n+1})})=\sum_{I\in{[n]\choose
2k-1}}\\#\\{\text{alternating permutations on $I\cup\\{n+1\\}$, whose $1$st
entry is $n+1$}\\}.$

In particular, when $n+1=2k$,
$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}(P_{2k})})$, which is known to be the
$k$th Catalan number [11], represents the number of $312$-avoiding alternating
permutations on $[n+1]=[2k]$. This provides an alternative proof that the
number of $312$-avoiding alternating permutations of length $2k$ is equal to
the $k$th Catalan number [40].

Hochschild polytopes, including Stanley-Pitman polytopes, are chordal
nestohedra, but most of them are not graph associahedra, as will be shown in
Propositions 2.5 and 2.7. For each pair $(m,n)$ of non-negative integers, the
$(m,n)$-Hochschild polytope $\mathds{H}\mathrm{och}(m,n)$ is defined as the
nestohedron associated with the building set $\mathcal{B}$ on a set
$[m+n]=\\{1,2,\ldots,m+n\\}$, where the building set $\mathcal{B}$ consists of
$I\subset[m+n]$ that satisfy the following condition:

$\left|I\right|\geq 2\Rightarrow I\cap[m+1,m+n]\text{ is either
}\emptyset\text{ or }[m+r,m+n]\text{ for }1\leq r\leq n,$

where $[i,j]\coloneqq\\{\ell\in\mathbb{Z}\colon i\leq\ell\leq j\\}$ for
integers $i\leq j$. See Section 2.3 for more details.

Although the complex and real toric manifolds corresponding to a Hochschild
polytope appear interesting, the authors are not aware of any existing
topological and algebraic geometrical studies on them as of the writing of
this paper. It would be appropriate to refer to such complex and real toric
manifolds as the _Hochschild_ and _real Hochschild variety_ , respectively.
Due to our main theorem, we are able to compute the Betti numbers of real
Hochschild varieties.

For non-negative integers $s$ and $r$, consider the set of alternating
permutations $(x_{1}x_{2}\cdots x_{s+r})$ on $[s+r]$ such that

$s+r\geq x_{i}>x_{j}\geq s+1\Rightarrow i<j,$

which is denoted by $\mathrm{Alt}_{\mathds{H}\mathrm{och}(s,r)}$.

######  Corollary 1.3.

For each non-negative integers $m$ and $n$, the $k$th Betti numbers of a real
Hochschild variety $X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)}$ are

$\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)})=\sum_{s+r=2k}{m\choose
s}\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(s,r)}\right|,$

where $s$ and $r$ are non-negative integers.

Moreover, if $n\geq m+2$,

$\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)})=\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,m+2)}),$

for all $k\geq 0$.

In the last section, we propose some interesting open problems related to the
nested set complex and toric topology, motivated by the results of this paper.

##  2\. Preliminaries

###  2.1. Real toric manifolds associated with nestohedra

Let us consider a complex $n$-dimensional toric manifold, denoted by $X$, and
its associated real toric manifold, denoted by $X^{\mathbb{R}}$. By the
fundamental theorem of toric geometry, $X$ corresponds to a non-singular
complete fan $\Sigma$ in $\mathbb{R}^{n}$. Since $\Sigma$ is a simplicial fan,
one can replace it with the pair $(K,\lambda)$ obtained by

  1. (1)

a simplicial complex $K$ on the set $V$ of rays in $\Sigma$ such that a
$(k-1)$-simplex of $K$ corresponds to a $k$-dimensional cone of $\Sigma$, and

  2. (2)

a linear map $\lambda$ from $V$ to $\mathbb{Z}^{n}$, called a _characteristic
map_ of $X$, such that the image $\lambda(v)$ is the primitive vector in the
direction of $v$ for each $v\in V$.

Then $X$ is completely determined by $(K,\lambda)$. In a similarly way,
$X^{\mathbb{R}}$ is completely determined by a pair
$(K,\lambda^{\mathbb{R}})$, where $\lambda^{\mathbb{R}}$ is the map, called a
_mod $2$ characteristic map_ of $X^{\mathbb{R}}$, defined by the composition
$V\xrightarrow{\lambda}\mathbb{Z}^{n}\xrightarrow{\text{mod}2}\mathbb{Z}_{2}^{n}$.
One may regard $\lambda^{\mathbb{R}}$ by an $n\times m$
$\mathbb{Z}_{2}$-matrix, where each column is $\lambda^{\mathbb{R}}(v)$ for
$v\in V$, and the row space of this matrix is denoted by
$\operatorname{row}\lambda^{\mathbb{R}}$. For each element
$\omega\in\operatorname{row}\lambda^{\mathbb{R}}$, $\omega$ can be regarded as
a subset of $V$ via the standard correlation between the power set of $V$ and
$\mathbb{Z}_{2}^{m}$. Hence, $\omega$ is associated with the induced
subcomplex $K_{\omega}$ of $K$ with respect to $\omega$.

Remark that the Betti numbers of $X^{\mathbb{R}}$ is determined by
$(K,\lambda^{\mathbb{R}})$ in accordance with the following theorem.

######  Theorem 2.1 ([43],[12]).

Let $X^{\mathbb{R}}$ be the real toric manifold associated with
$(K,\lambda^{\mathbb{R}})$. Then the $k$th rational Betti number is given by

$\beta_{k}(X^{\mathbb{R}})=\sum_{\omega\in\operatorname{row}\lambda^{\mathbb{R}}}\widetilde{\beta}_{k-1}(K_{\omega}),$

where $K_{\omega}$ is the induced subcomplex of $K$ with respect to $\omega$
and $\widetilde{\beta}_{k-1}$ denotes the $(k-1)$th reduced Betti number.

For a finite set $S$ of natural numbers, a finite collection $\mathcal{B}$ of
nonempty subsets of $S$ is a _building set_ on $S$ if

  1. (1)

$\\{i\\}\in\mathcal{B}$ for $i\in S$, and

  2. (2)

if $I,J\in\mathcal{B}$ and $I\cap J\neq\emptyset$, then $I\cup
J\in\mathcal{B}$.

An inclusion-maximal subset in $\mathcal{B}$ is called a _connected component_
of $\mathcal{B}$, and we call $\mathcal{B}$ a _connected building set_ if
$S\in\mathcal{B}$. For a building set $\mathcal{B}$ on $S$ with $I\subset S$,
the _restricted building set_ $\mathcal{B}|_{I}$ of $\mathcal{B}$ to $I$ is
$\\{J\in\mathcal{B}\colon J\subset I\\}$.

Let $\mathcal{B}$ be a connected building set on $[n+1]=\\{1,\ldots,n+1\\}$.
The _nestohedron_ $P_{\mathcal{B}}$ with respect to $\mathcal{B}$ is defined
as the Minkowski sum of simplices

$P_{\mathcal{B}}=\sum_{I\in\mathcal{B}}{\operatorname{conv}(I)},$

where $\operatorname{conv}(I)$ is the convex hull of the $i$th coordinate
vectors in $\mathbb{R}^{n+1}$ for all $i\in I$. Denote by $K_{\mathcal{B}}$
the face structure of $P_{\mathcal{B}}$ with reverse inclusion. Then, since
each facet of $P_{\mathcal{B}}$ can be identified with an element of
$\mathcal{B}\setminus\\{[n+1]\\}$, $K_{\mathcal{B}}$ can be regarded as a
simplicial complex on $\mathcal{B}\setminus\\{[n+1]\\}$. A subset
$N\subseteq\mathcal{B}\setminus\\{[n+1]\\}$ is called a _nested set_ if it
satisfies the following conditions:

  1. (1)

For each $I,J\in N$ one has either $I\subseteq J$, $J\subseteq I$, or $I\cap
J=\emptyset$.

  2. (2)

For any collection of $k\geq 2$ disjoint subsets $I_{1},\ldots,I_{k}\in N$,
the union $I_{1}\cup\cdots\cup I_{k}\notin\mathcal{B}$.

According to [35, Theorem 7.4], $K_{\mathcal{B}}$ is isomorphic to the
simplicial complex on $\mathcal{B}\setminus\\{[n+1]\\}$ which is defined by
the set of all nested sets of $\mathcal{B}$, known as a _nested set complex_ ,
as abstract simplicial complexes.

In addition, since the dimension of $P_{\mathcal{B}}$ is $n$, one can define
maps
$\lambda_{\mathcal{B}}\colon\mathcal{B}\setminus\\{[n+1]\\}\to\mathbb{Z}^{n}$
and
$\lambda_{\mathcal{B}}^{\mathbb{R}}\colon\mathcal{B}\setminus\\{[n+1]\\}\to\mathbb{Z}_{2}^{n}$
by $\lambda(I)$ is the primitive vector in the direction of the outward normal
vector of the facet corresponding to $I$, and
$\lambda^{\mathbb{R}}(I)\equiv\lambda(I)\pmod{2}$ for all
$I\in\mathcal{B}\setminus\\{[n+1]\\}$. More precisely,
$\lambda^{\mathbb{R}}_{\mathcal{B}}$ is given by

$\lambda^{\mathbb{R}}_{\mathcal{B}}(I)=\sum_{i\in I}e_{i},$

where $e_{i}$ is the $i$th standard vector of $\mathbb{Z}^{n}_{2}$ for $1\leq
i\leq n$ and $e_{n+1}=\sum_{i=1}^{n}e_{i}$.

By [35, Proposition 7.10], the pair $(K_{\mathcal{B}},\lambda_{\mathcal{B}})$
determines the toric manifold, denoted by $X_{\mathcal{B}}$, and the pair
$(K_{\mathcal{B}},\lambda^{\mathbb{R}}_{\mathcal{B}})$ determines the real
toric manifold, denoted by $X^{\mathbb{R}}_{\mathcal{B}}$. The main purpose of
this paper is to compute the rational Betti numbers of
$X^{\mathbb{R}}_{\mathcal{B}}$ using Theorem 2.1.

Let $\omega_{i}$ be the element of
$\operatorname{row}\lambda_{\mathcal{B}}^{\mathbb{R}}$ corresponding to the
$i$th row of $\lambda_{\mathcal{B}}^{\mathbb{R}}$ as a matrix for
$i=1,\ldots,n$. Then, each element
$\omega=\omega_{i_{1}}+\cdots+\omega_{i_{k}}$ of
$\operatorname{row}\lambda_{\mathcal{B}}^{\mathbb{R}}$ can be identified with
a subset $I_{\omega}\subset[n+1]$ that has an even cardinality as follows:

(2.1) $I_{\omega}=\begin{cases}\\{i_{1},\ldots,i_{k}\\},&\mbox{if }k\mbox{ is
even}\\\ \\{i_{1},\ldots,i_{k},n+1\\},&\mbox{otherwise}.\end{cases}$

For an even cardinality subset $I\subset[n+1]$, denote by
$(K_{\mathcal{B}})_{I}$ the full subcomplex of $K_{\mathcal{B}}$ induced by
all vertices $J$, where $|J\cap I|$ is odd. Then, one can easily see that
$(K_{\mathcal{B}})_{\omega}$ is equal to $(K_{\mathcal{B}})_{I_{\omega}}$,
and, conversely, for each an even subset $I$ of $[n+1]$, there is an element
$\omega\in\operatorname{row}\lambda_{\mathcal{B}}^{\mathbb{R}}$ corresponding
to $I$. Therefore, for a connected building set $\mathcal{B}$ on $[n+1]$, the
Betti number formula in Theorem 2.1 can be written as

(2.2)
$\beta_{k}(X_{\mathcal{B}}^{\mathbb{R}})=\sum_{\begin{subarray}{c}I\subset[n+1]\\\
\left|I\right|\text{ is
even}\end{subarray}}\widetilde{\beta}_{k-1}((K_{\mathcal{B}})_{I}).$

In this paper, we are particularly interested in some special class of
building sets. A building set $\mathcal{B}$ on $S$ is _chordal_ if for any
$I=\\{i_{1}<\cdots<i_{r}\\}\in\mathcal{B}$, $\\{i_{s},\ldots,i_{r}\\}$ is also
an element of $\mathcal{B}$ for each $1<s<r$. It can be immediately observed
that every restricted building set of a chordal building set is also chordal.
A nestohedron $P_{\mathcal{B}}$ is called a _chordal nestohedron_ if
$\mathcal{B}$ is chordal.

######  Example 2.2.

A building set

$\mathcal{B}=\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{1,4\\},\\{3,4\\},\\{1,3,4\\},\\{2,3,4\\},\\{1,2,3,4\\}\\}$

is an example of connected chordal building set on $\\{1,2,3,4\\}$. The
chordal nestohedron $P_{\mathcal{B}}$ is illustrated as in Figure 2, and each
facet of $P_{\mathcal{B}}$ is labeled by the corresponding element of
$\mathcal{B}$.

$\\{1,4\\}$$\\{3,4\\}$$\\{1,3,4\\}$$\\{1\\}$$\\{3\\}$$\\{2\\}$$\\{2,3,4\\}$$\\{4\\}$
Figure 2. Chordal nestohedron $P_{\mathcal{B}}$

Let $K_{\mathcal{B}}$ be the face structure of $P_{\mathcal{B}}$ with reverse
inclusion, and $\lambda_{\mathcal{B}}^{\mathbb{R}}$ its mod $2$ characteristic
map. Then, $\lambda_{\mathcal{B}}^{\mathbb{R}}$ is represented by the $3\times
8$ matrix

$\lambda_{\mathcal{B}}^{\mathbb{R}}=\left(\begin{array}[]{ccccccccc}\\{1\\}&\\{2\\}&\\{3\\}&\\{4\\}&\\{1,4\\}&\\{3,4\\}&\\{1,3,4\\}&\\{2,3,4\\}\\\
\hline\cr 1&0&0&1&0&1&0&1\\\ 0&1&0&1&1&1&1&0\\\ 0&0&1&1&1&0&0&0\\\
\end{array}\right),$

where the first row indicates the labels of each facet. Let $\omega_{i}$ be
the $i$th row of $\operatorname{row}\lambda_{\mathcal{B}}^{\mathbb{R}}$ for
$1\leq i\leq 3$. For instance, if we take $\omega=\omega_{1}+\omega_{2}$, the
corresponding even subset is $I_{\omega}=\\{1,2\\}$. One can see that both
$(K_{\mathcal{B}})_{\omega_{1}+\omega_{2}}$ and
$(K_{\mathcal{B}})_{\\{1,2\\}}$ are the full subcomplex induced by the set

$\\{J\in\mathcal{B}\colon|J\cap\\{1,2\\}|\equiv
1\pmod{2}\\}=\\{\\{1\\},\\{2\\},\\{1,4\\},\\{1,3,4\\},\\{2,3,4\\}\\}.$

In this case, $(K_{\mathcal{B}})_{I_{\omega}}$ is contractible, and, hence,
$\tilde{\beta}_{\ast}((K_{\mathcal{B}})_{I_{\omega}})$ vanishes. Therefore, it
contribute nothing to the Betti number of $X_{\mathcal{B}}^{\mathbb{R}}$. If
we take $\omega=\omega_{1}+\omega_{2}+\omega_{3}$, the corresponding even
subset is $I_{\omega}=\\{1,2,3,4\\}$. One can see that both
$(K_{\mathcal{B}})_{\omega}$ and $(K_{\mathcal{B}})_{I_{\omega}}$ are the full
subcomplex induced by the set

$\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{1,3,4\\},\\{2,3,4\\}\\}$

which is homotopy equivalent to a circle $S^{1}$. Therefore, it contributes
one to $\beta_{2}(X_{\mathcal{B}}^{\mathbb{R}})$.

From the above example and many other similar examples, one can find
interesting patterns in the computation of the reduced Betti numbers of
$(K_{\mathcal{B}})_{I}$; if $|I|=2k$, then $(K_{\mathcal{B}})_{I}$ is always
homotopy equivalent to a bouquet of $(k-1)$-dimensional spheres, so it
contributes only to $\beta_{k}(X_{\mathcal{B}}^{\mathbb{R}})$. The purposes of
this paper is to show that this phenomenon is not just a coincidence of
specific cases, but rather a remarkable fact that holds generally for chordal
building sets. Refer Section 3. In addition, we also show in Section 4 that
the number of spheres in each bouquet is obtainable by counting permutations
satisfying conditions dependent on $\mathcal{B}$.

###  2.2. Graph associahedra

A building set $\mathcal{B}$ on $S$ is said to be _graphical_ if there is a
finite simple graph $G$ on $S$ such that $I\in\mathcal{B}$ if and only if the
induced subgraph $G|_{I}$ on $I$ is connected. In this case, $\mathcal{B}$ is
denoted by $\mathcal{B}(G)$. Furthermore, every restricted building set
$\mathcal{B}(G)|_{I}$ to $I$ is exactly $\mathcal{B}(G|_{I})$, where $G|_{I}$
is the induced subgraph on $I$. For a finite simple graph $G$, the nestohedron
$P_{\mathcal{B}(G)}$ is called a _graph associahedron_. Graph associahedra
have been widely studied in [5], [35], [45], and [46].

The Betti number of the real toric manifolds corresponding to a graphical
building set is well-studied in [11]. For a simple graph $G$ on $S$, the _$a$
-number_ $a(G)$ of $G$ is defined as follows:

  * •

$a(G)=sa(G)=1$.

  * •

If $G$ is connected,

$sa(G)=\begin{cases}-\underset{I\subsetneq S}{\sum}sa(G|_{I}),&\mbox{if
$\left|S\right|$ is even,}\\\ 0,&\mbox{otherwise,}\end{cases}$

and $a(G)=\left|sa(G)\right|$.

  * •

$a(G)$ is the product of $a(G_{i})$ for $1\leq i\leq l$, where
$G_{1},\ldots,G_{l}$ are the connected components of $G$.

By [11, Theorem 1.1], the $k$th Betti numbers of the associated real toric
manifold $X^{\mathbb{R}}_{\mathcal{B}(G)}$ is given by

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}(G)})=\sum_{I\subset{S\choose
2k}}a(G|_{I}).$

A finite simple graph $G$ on a set $S$ is _chordal_ if $G$ contains no induced
$k$-cycle for $k\geq 4$. Refer [3] for additional details. The building set of
a chordal graph is not necessary to be chordal. However, if we give an
appropriate labeling to nodes of a chordal graph, we can obtain a chordal
building set. A _perfect elimination ordering_ in $G$ is an ordering of the
vertices of $G$ such that $G$ has no induced subgraph $G|_{\\{i,j,k\\}}$ with
edges $(i,j),(i,k)$ but without the edge $(j,k)$, where $i<j<k$. A graph $G$
is chordal if and only if $G$ admits a perfect elimination ordering [21].
According to [36, Proposition 9.4], $G$ is a chordal graph labeled by a
perfect elimination ordering if and only if the graphical building set
$\mathcal{B}(G)$ is a chordal building set.

######  Example 2.3.

Consider a path graph $P_{4}$ with $4$ vertices, which is a chordal graph. If
$P_{4}$ is labeled by

$2$$3$$1$$4$

,

then the graphical building set $\mathcal{B}(P_{4})$ of $P_{4}$

$\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{2,3\\},\\{1,3\\},\\{1,4\\},\\{1,2,3\\},\\{1,3,4\\},\\{1,2,3,4\\}\\}$

is not chordal. However, if we label $P_{4}$ by

$1$$2$$3$$4$

that gives a perfect elimination ordering, then $\mathcal{B}(P_{4})$

$\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{1,2,3\\},\\{2,3,4\\},\\{1,2,3,4\\}\\}$

is chordal.

###  2.3. Hochschild polytopes

For non-negative integers $m$ and $n$, the _$(m,n)$ -Hochschild polytope_
$\mathds{H}\mathrm{och}(m,n)$ was introduced in [34]. To describe its
definition, we briefly introduce some notions following:

  * •

A finite sequence $\mathbb{S}=\\{(s_{1},c_{1}),\ldots,(s_{\ell},c_{\ell})\\}$
is called an _$m$ -lighted $n$-shade_ if

    1. (1)

$s_{i}$ is a (possibly empty) tuple of positive integers and has total sum
$n$,

    2. (2)

$c_{1},\ldots,c_{\ell}$ are pairwise disjoint subsets of $[m]$ that their
union $c_{1}\cup c_{2}\cup\cdots\cup c_{\ell}$ is $[m]$.

    3. (3)

$s_{i}$ and $c_{i}$ cannot be both empty for each $i=1,\ldots,\ell$.

  * •

A _rank_ of $\mathbb{S}=\\{(s_{1},c_{1}),\ldots,(s_{\ell},c_{\ell})\\}$ is
defined by $m-\ell+\sum_{i=1}^{\ell}\left|s_{i}\right|$.

  * •

For $m$-lighted $n$-shades
$\mathbb{S}=\\{(s_{1},c_{1}),\ldots,(s_{\ell},c_{\ell})\\}$ and
$\mathbb{S}^{\prime}$, we denote by $\mathbb{S}^{\prime}\to\mathbb{S}$ if
$\mathbb{S}$ and $\mathbb{S}^{\prime}$ satisfy one of the following:

    1. (1)

there is $1\leq i<\ell$ such that $\mathbb{S}^{\prime}$ is obtained from
$\mathbb{S}$ by exchange $(s_{i},c_{i}),(s_{i+1},c_{i+1})$ to
$(s_{i}+s_{i+1},c_{i}\cup c_{i+1})$, where
$s_{i}=(s_{i}^{1},\ldots,s_{i}^{p}),s_{i+1}=(s_{i+1}^{1},\ldots,s_{i+1}^{q})$,
and
$s_{i}+s_{i+1}=(s_{i}^{1},\ldots,s_{i}^{p},s_{i+1}^{1},\ldots,s_{i+1}^{q})$.

    2. (2)

there is $1\leq i\leq\ell$ such that $\mathbb{S}^{\prime}$ is obtained from
$\mathbb{S}$ by exchange $(s_{i},c_{i})$ to $(s^{\prime}_{i},c_{i})$, where
$s_{i}=(s_{i}^{1},\ldots,s_{i}^{p})$,
$s^{\prime}_{i}=(s_{i}^{1},\ldots,s_{i}^{r_{1}},s_{i}^{r_{2}},\ldots,s_{i}^{p})$,
and $s_{i}^{r}=s_{i}^{r_{1}}+s_{i}^{r_{2}}$ for some $1\leq r\leq p$.

  * •

$(\mathcal{P}_{(m,n)},\leq)$ is the poset of $m$-lighted $n$-shades with the
partial order $\leq$ defined by

$\mathbb{S}^{\prime}\leq\mathbb{S}\Leftrightarrow\begin{cases}\mathbb{S}^{\prime}\to\mathbb{S}_{1}\to\cdots\to\mathbb{S},\\\
\mathbb{S}^{\prime}=\mathbb{S}.\end{cases}$

For an $m$-lighted $n$-shade
$\mathbb{S}=\\{(s_{1},c_{1}),\ldots,(s_{\ell},c_{\ell})\\}$, $\mathbb{S}$ is
visually represented as a vertical line with the tuples
$s_{1},\ldots,s_{\ell}$ on the left, and the sets of $c_{1},\ldots,c_{\ell}$
on the right. Since there is no confusion, we use the notation $a_{1}\cdots
a_{k}$ for the tuple $(a_{1},\ldots,a_{k})$ or the set
$\\{a_{1},\ldots,a_{k}\\}$. See Figure 3 for examples.

121111rank 5$\to$211111rank 4$\to$21211rank 3$\to$21211rank 2$\to$21211rank
1$\to$21211rank 0 Figure 3. A maximal chain of $2$-lighted $4$-shades

######  Remark 2.4.

The poset of $1$-lighted $n$-shades gives a realization of the Hochschild
lattice, which were introduced in [6]. Because of nice lattice properties the
Hochschild lattice, it has been studied in combinatorics [15] and [30].

For non-negative integers $m$ and $n$, the $(m,n)$-Hochschild polytope
$\mathds{H}\mathrm{och}(m,n)$ is defined as an abstract polytope whose face
poset is anti-isomorphic to $(\mathcal{P}_{(m,n)},\leq)$. Refer [34] for the
well-definedness of $\mathds{H}\mathrm{och}(m,n)$. As brief examples, see
Figure 4 for $(m,n)$-Hochschild polytopes and their face posets, represented
by $m$-lighted $n$-shades for $m+n=3$.

123123123213213132132213123132312231123121121121211211112112211121112112211121111111111111111111111121211112111111111212131211112
Figure 4. $\mathds{H}\mathrm{och}(m,n)$ for $(m,n)=(3,0),(2,1),(1,2)$ and
$(0,3)$

Let $\mathcal{B}_{m,n}$ be the set consisting of non-empty subsets $I$ of
$[m+n]$ such that

(2.3) $\left|I\right|\geq 2\Rightarrow I\cap[m+1,m+n]\text{ is either
}\emptyset\text{ or }[m+r,m+n]\text{ for }1\leq r\leq n,$

where $[i,j]$ denotes the set of all integers $\ell$ such that $i\leq\ell\leq
j$. It is clear that $\mathcal{B}_{m,n}$ is a connected chordal building set
on $[m+n]$.

######  Proposition 2.5.

For non-negative integers $m$ and $n$, the nestohedron $P_{\mathcal{B}_{m,n}}$
has a same combinatorial structure as the $(m,n)$-Hochschild polytope
$\mathds{H}\mathrm{och}(m,n)$.

###### Proof.

Consider an $m$-lighted $n$-shade

$\mathbb{S}=\\{(s_{1},c_{1}),\ldots,(s_{\ell}=(s^{1}_{\ell},\ldots,s^{p}_{\ell}),c_{\ell})\\},$

whose rank is not $m+n-1$. When there exists an index $1\leq i\leq p$ such
that $s^{i}_{\ell}$ is at least $2$, we denote the maximal such index by
$1\leq q\leq p$. Define a subset

$B_{\mathbb{S}}=\begin{cases}c_{\ell}\cup\\{m+n-i\colon 0\leq
i<\left|s_{\ell}\right|\\},&\mbox{if all entries of $s_{\ell}$ are $1$},\\\
\\{m+n-(p+1-q)\\},&\mbox{otherwise,}\end{cases}$

of $[m+n]$, and an $m$-lighted $n$-shade $\varphi(\mathbb{S})$ obtained from
$\mathbb{S}$ as follows:

  1. (1)

By exchange $(s_{\ell-1},c_{\ell-1}),(s_{\ell},c_{\ell})$ to
$(s_{\ell-1}+s_{\ell},c_{\ell-1}\cup c_{\ell})$ if all entries of $s_{\ell}$
are $1$,

  2. (2)

By exchange $s_{\ell}=(\ldots,s^{q}_{\ell},\ldots)$ to
$(\ldots,s_{\ell}^{q}-1,1,\ldots)$, otherwise.

For any non-negative integer $i$, $\varphi^{i}(\mathbb{S})$ is denoted by the
composition

$\varphi^{i}(\mathbb{S})=\begin{cases}\mathbb{S}&\mbox{if }i=0\\\
\underbrace{\varphi\cdot\varphi\cdot\cdots\cdot\varphi}_{\text{$i$
times}}(\mathbb{S}),&\mbox{otherwise}.\end{cases}$

For an $m$-lighted $n$-shade $\mathbb{S}$ whose rank is $r$, define

$\Phi(\mathbb{S})=\\{B_{\varphi^{i}(\mathbb{S})}\colon 0\leq i\leq
m+n-2-r\\}.$

Indeed, $\Phi$ induces a well-defined order isomorphism from
$(\mathcal{P}_{(m,n)}\setminus\\{[m+n]\\},\leq)$ to the nested set complex
$K_{\mathcal{B}_{m,n}}$. ∎

######  Example 2.6.

Let $\mathcal{C}$ be a chain of $2$-lighted $4$-shades as follows:

21211rank 3$\leq$21211rank 1$\leq$21211rank 0

Then the chain $\mathcal{C}$ corresponds to the chain of nested sets

$\\{\\{1,6\\},\\{4\\}\\}\subset\\{\\{6\\},\\{1,6\\},\\{4\\},\\{1,4,5,6\\}\\}\subset\\{\\{6\\},\\{1,6\\},\\{4\\},\\{1,4,5,6\\},\\{1,3,4,5,6\\}\\}$

of $\mathcal{B}_{m,n}\setminus\\{[m+n]\\}$.

For each pair $(m,n)$ of non-negative integers, since the $(m,n)$-Hochschild
polytope can be realized as a nestohedron $P_{\mathcal{B}_{m,n}}$,
$\mathds{H}\mathrm{och}(m,n)$ induces the complex and real toric manifold, and
they are denoted by $X_{\mathds{H}\mathrm{och}(m,n)}$ and
$X_{\mathds{H}\mathrm{och}(m,n)}^{\mathbb{R}}$, respectively. Moreover, we
call such complex and real toric manifolds _Hochschild_ and _real Hochschild
variety_ , respectively.

In particular, $\mathds{H}\mathrm{och}(0,n)$ is known as a Stanley-Pitman
polytope [40]. It is easy to verify that $\mathds{H}\mathrm{och}(m,n)$ forms a
graph associahedron when $n\leq 2$, and specifically, it forms a permutohedron
when $n\leq 1$.

######  Proposition 2.7.

For $n\geq 3$, a building set $\mathcal{B}_{m,n}$ is not a graphical building
set.

###### Proof.

Assume that $\mathcal{B}=\mathcal{B}_{m,n}$ is graphical when $n\geq 3$.
Choose a finite simple graph $G$ with $\mathcal{B}=\mathcal{B}(G)$. Let
$I=\\{m+1,\ldots,m+n\\}$. Then the restricted building set
$\mathcal{B}_{m,n}|_{I}$ should be a building set with respect to an subgraph
of $G$ induced by $I$. Since $I\in\mathcal{B}_{m,n}|_{I}$, the subgraph is
connected, and it consists of at least $n-1$ edges. However, it contradicts
that $\mathcal{B}_{m,n}|_{I}$ has a unique element of cardinality $2$. ∎

##  3\. EL-shellability of the nested set complex

For a graded poset $\mathcal{P}$, let us consider the _order complex_
$\Delta(\mathcal{P})$, that is, $\Delta(\mathcal{P})$ is the simplicial
complex whose simplices are finite chains of $\mathcal{P}$. We assume that a
graded poset $\mathcal{P}$ is _bounded_ , which means that $\mathcal{P}$ has
both a least and a greatest element.

Let $\mathbf{E}(\mathcal{P})$ denote the set of edges of the Hasse diagram of
$\mathcal{P}$, which is the set of pairs of elements in $\mathcal{P}$ such
that one covers the other. An _edge labeling_ $\mu$ of $\mathcal{P}$ is a map
from $\mathbf{E}(\mathcal{P})$ to a poset $(\mathcal{Q},\succeq)$. For an edge
labeling $\mu$ and for each chain $c=(z_{0},z_{1},\dots,z_{k})$ of
$\mathcal{P}$ such that $z_{i}$ covers $z_{i-1}$ for all $i=1,\ldots,k$, we
denote by $\mu(c)$ the $k$-tuple

$\mu(c)=(\mu(z_{0},z_{1}),\mu(z_{1},z_{2}),\ldots,\mu(z_{k-1},z_{k})).$

An edge labeling $\mu\colon\mathbf{E}(\mathcal{P})\to\mathcal{Q}$ is _reversed
edge lexicographical_ if for each interval $[x,y]$ in $\mathcal{P}$,

  1. (1)

there is a unique maximal chain $c=(z_{0},z_{1},\dots,z_{k})$ of $[x,y]$ such
that $\mu(c)$ is _decreasing_ , i.e.

$\mu(z_{0},z_{1})\succeq\mu(z_{1},z_{2})\succeq\cdots\succeq\mu(z_{k-1},z_{k}),\text{
and}$

  2. (2)

for all other maximal chains
$c^{\prime}=(z_{0}^{\prime},\dots,z_{k}^{\prime})$ of $[x,y]$, $\mu(c)$
lexicographically succeeds $\mu(c^{\prime})$, i.e.
$\mu(z_{i-1},z_{i})\succ\mu(z_{i-1}^{\prime},z_{i}^{\prime})$ for
$i=\min\\{1\leq j\leq
k\colon\mu(z_{j-1},z_{j})\neq\mu(z_{j-1}^{\prime},z_{j}^{\prime})\\}$.

A bounded graded poset $\mathcal{P}$ is said to be _EL-shellable_ if
$\mathcal{P}$ admits an reversed edge lexicographical edge labeling. 111The
original definition of EL-shellability in [1] requires for $\mathcal{P}$ to
admit an edge labeling $\mu$ that is edge lexicographical. However, in this
case, $-\mu$ is a reversed edge lexicographical edge labeling, making our
definition essentially equivalent to the original one.

In this section, unless otherwise stated, we consider a set of natural numbers
with an even cardinality $2k$. For a building set $\mathcal{B}$ on $S$, define
the set $\widehat{\mathcal{P}}_{\mathcal{B}}$ of subsets $I$ of $S$ such that

$\widehat{\mathcal{P}}_{\mathcal{B}}\coloneqq\\{I\subset
S\colon\text{$\mathcal{B}|_{I}$ has no connected components of odd order}\\}.$

Note that $(\widehat{\mathcal{P}}_{\mathcal{B}},\subseteq)$ is a bounded
graded poset, and each edge $(I_{1},I_{2})$ of
$\mathbf{E}(\widehat{\mathcal{P}}_{\mathcal{B}})$ consists of two subsets
satisfying $I_{1}\subset I_{2}$ and $\left|I_{2}\right|=\left|I_{1}\right|+2$.
We note that $I_{2}\setminus I_{1}$ should be contained in the same connected
component, denoted by $\mathsf{C}(I_{2};I_{1})$, of the restricted building
set $\mathcal{B}|_{I_{2}}$.

Now, we introduce some notions to define an edge labeling on
$\widehat{\mathcal{P}}_{\mathcal{B}}$ that will be reversed edge
lexicographical.

######  Definition 3.1.

For the set $\Omega=\mathbb{Z}\times\mathbb{Z}$, let $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ be the subsets of $\Omega\times\Omega$ defined by

$\displaystyle\mathcal{R}_{1}=\\{((x_{1},x_{2}),(y_{1},y_{2}))\in\Omega\times\Omega\colon
x_{1}\geq x_{2}\geq y_{1}\geq y_{2}\\},$
$\displaystyle\mathcal{R}_{2}=\\{((x_{1},x_{2}),(y_{1},y_{2}))\in\Omega\times\Omega\colon
x_{1}=y_{1}\geq x_{2}\geq y_{2}\\}.$

For each pair of elements $\alpha=(\alpha_{1},\alpha_{2})$ and
$\beta=(\beta_{1},\beta_{2})$ of the set $\Omega$, define the relation $\geq$
on $\Omega$ by $\alpha\geq\beta$ if the pair $(\alpha,\beta)$ is an element of
either $\mathcal{R}_{1}$ or $\mathcal{R}_{2}$.

######  Proposition 3.2.

The relation $\geq$ is a partial order on $\mathbb{Z}\times\mathbb{Z}$.

###### Proof.

To begin with, for each $\alpha\in\Omega$, since
$(\alpha,\alpha)\in\mathcal{R}_{2}$, the reflexivity holds.

For each pair of elements $\alpha=(\alpha_{1},\alpha_{2})$ and
$\beta=(\beta_{1},\beta_{2})$ of $\Omega$, if $\alpha\geq\beta$ and
$\beta\geq\alpha$, then

$\begin{cases}\alpha_{1}\geq\alpha_{2}\geq\beta_{1}\geq\beta_{2}\geq\alpha_{1}>\alpha_{2},&\mbox{if
}(\alpha,\beta),(\beta,\alpha)\in\mathcal{R}_{1},\\\
\alpha_{1}\geq\alpha_{2}\geq\beta_{1}=\alpha_{1}\geq\beta_{2}\geq\alpha_{2},&\mbox{if
}(\alpha,\beta)\in\mathcal{R}_{1},(\beta,\alpha)\in\mathcal{R}_{2},\\\
\alpha_{1}=\beta_{1}\geq\alpha_{2}\geq\beta_{2}\geq\alpha_{2},&\mbox{if
}(\alpha,\beta),(\beta,\alpha)\in\mathcal{R}_{2}.\end{cases}$

Consequently, in all cases, we conclude that $\alpha=\beta$ which confirms the
antisymmetry.

Let $\alpha=(\alpha_{1},\alpha_{2})$, $\beta=(\beta_{1},\beta_{2})$, and
$\gamma=(\gamma_{1},\gamma_{2})$ be elements of $\Omega$. Assume that
$\alpha\geq\beta$ and $\beta\geq\gamma$. Then, we have the following four
cases.

$\begin{cases}\alpha_{1}\geq\alpha_{2}\geq\beta_{1}\geq\beta_{2}\geq\gamma_{1}\geq\gamma_{2},&\mbox{if
}(\alpha,\beta),(\beta,\gamma)\in\mathcal{R}_{1},\\\
\alpha_{1}\geq\alpha_{2}\geq\beta_{1}=\gamma_{1}\geq\beta_{2}\geq\gamma_{2},&\mbox{if
}(\alpha,\beta)\in\mathcal{R}_{1},(\beta,\gamma)\in\mathcal{R}_{2},\\\
\alpha_{1}=\beta_{1}\geq\alpha_{2}\geq\beta_{2}\geq\gamma_{1}\geq\gamma_{2},&\mbox{if
}(\alpha,\beta)\in\mathcal{R}_{2},(\beta,\gamma)\in\mathcal{R}_{1},\\\
\alpha_{1}=\beta_{1}=\gamma_{1}\geq\alpha_{2}\geq\beta_{2}\geq\gamma_{2},&\mbox{if
}(\alpha,\beta),(\beta,\gamma)\in\mathcal{R}_{2}.\end{cases}$

In the first three cases, $(\alpha,\gamma)\in\mathcal{R}_{1}$, and, in the
final case, $(\alpha,\gamma)\in\mathcal{R}_{2}$. Therefore,
$\alpha\geq\gamma$, which proves the transitivity. ∎

Let $\succeq$ be the lexicographic order on the product
$(\mathbb{Z},\geq)\times(\Omega,\geq)$, that is,

$(x,\alpha)\succeq(y,\beta)\text{ if and only if }\begin{cases}x>y,\text{
or}\\\ x=y,\alpha\geq\beta.\end{cases}$

Define an edge labeling
$\mu_{\mathcal{B}}\colon\mathbf{E}(\widehat{\mathcal{P}}_{\mathcal{B}})\to(\mathbb{Z}\times\Omega,\succeq)$
by

$\mu_{\mathcal{B}}(I_{1},I_{2})=(\max\mathsf{C}(I_{2};I_{1}),(\max
I_{2}\setminus I_{1},\min I_{2}\setminus I_{1}))$

for each $(I_{1},I_{2})\in\mathbf{E}(\widehat{\mathcal{P}}_{\mathcal{B}})$.

######  Example 3.3.

Consider the building set $\mathcal{B}=\mathcal{B}_{2,4}$. The Hasse diagram
$\mathbf{E}(\widehat{\mathcal{P}}_{\mathcal{B}})$ of $\mathcal{B}$ and the
edge labeling $\mu_{\mathcal{B}}$ are illustrated in Figure 5.

$\emptyset$$\\{2,6\\}$$\\{5,6\\}$$\\{1,2\\}$$\\{1,6\\}$$\\{2,4,5,6\\}$$\\{3,4,5,6\\}$$\\{1,4,5,6\\}$$\\{1,2,5,6\\}$$\\{1,2,3,4,5,6\\}$$(6,(6,5))$$(6,(6,2))$$(6,(6,1))$$(2,(2,1))$$(6,(5,4))$$(6,(5,2))$$(6,(4,2))$$(6,(4,3))$$(6,(4,1))$$(6,(2,1))$$(6,(6,5))$$(6,(5,1))$$(6,(5,4))$$(6,(3,1))$$(6,(2,1))$$(6,(3,2))$$(6,(4,3))$
Figure 5. $\mathbf{E}(\widehat{\mathcal{P}}_{\mathcal{B}_{2,4}})$ and
$\mu_{\mathcal{B}_{2,4}}$

For an interval $[\emptyset,\\{1,2,3,4,5,6\\}]$, the chain

$c=(\emptyset,\\{5,6\\},\\{3,4,5,6\\},\\{1,2,3,4,5,6\\})$

of $\widehat{\mathcal{P}}_{\mathcal{B}}$ is the unique maximal chain such that
$\mu_{\mathcal{B}}(c)$ is decreasing and, for all other maximal chains
$c^{\prime}$ of $\widehat{\mathcal{P}}_{\mathcal{B}}$, $\mu_{\mathcal{B}}(c)$
lexicographically succeeds $\mu_{\mathcal{B}}(c^{\prime})$. For other
intervals, one can find such a unique maximal chain. Therefore,
$\mu_{\mathcal{B}}$ forms a reversed lexicographical edge labeling, which
confirms that $\widehat{\mathcal{P}}_{\mathcal{B}_{2,4}}$ is EL-shellable.

The following lemma represents one of the central and most crucial ideas of
this research.

######  Lemma 3.4.

If $\mathcal{B}$ is a chordal building set, then
$(\widehat{\mathcal{P}}_{\mathcal{B}},\subseteq)$ is EL-shellable.

###### Proof.

Take any interval $[I,J]$ of $\widehat{\mathcal{P}}_{\mathcal{B}}$, where
$I\subsetneq J$. Let $\mathcal{C}_{1},\ldots,\mathcal{C}_{r}$ be all connected
components of the restricted building set $\mathcal{B}|_{J}$. We assume that
$\max\mathcal{C}_{1}>\max\mathcal{C}_{2}>\cdots>\max\mathcal{C}_{r}$ for all
$1\leq i\leq r$. Since both $\mathcal{B}|_{I}$ and $\mathcal{B}|_{J}$ have no
connected component of odd order, each cardinality of
$\mathcal{C}_{i}\setminus I$, denoted by $2k_{i}$, is even. Then the
cardinality of $J\setminus I$, denoted by $2N$, is $2k_{1}+\cdots+2k_{r}$. For
each $1\leq i\leq r$, we express

$\mathcal{C}_{i}\setminus
I=\\{\alpha_{2k_{0}+\cdots+2k_{i-1}+1},\alpha_{2k_{0}+\cdots+2k_{i-1}+2},\ldots,\alpha_{2k_{0}+\cdots+2k_{i-1}+2k_{i}}\\},$

where $k_{0}=0$ and
$\alpha_{2k_{0}+\cdots+2k_{i-1}+s}>\alpha_{2k_{0}+\cdots+2k_{i-1}+t}$ if
$1\leq s<t\leq 2k_{i}$.

Let $1\leq\ell\leq N$ be given. Define
$I_{\ell}=I\cup\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{2\ell}\\}$ and the
index $1\leq s_{\ell}\leq r$ such that
$\mathsf{C}(I_{\ell};I_{\ell-1})\subset\mathcal{C}_{s_{\ell}}$. Then one can
express

(3.1)
$I_{\ell}=I\cup\\{\alpha_{1},\ldots,\alpha_{2\ell}\\}=I\cup\mathcal{C}_{0}\cup\cdots\cup\mathcal{C}_{s_{\ell}-1}\cup\\{\alpha\in\mathcal{C}_{s_{\ell}}\colon\alpha\geq\alpha_{2\ell}\\},$

where $\mathcal{C}_{0}=\emptyset$. By the chordality of the building set
$\mathcal{B}$, the subset
$\\{\alpha\in\mathcal{C}_{s_{\ell}}\colon\alpha\geq\alpha_{2\ell}\\}$ of
$\mathcal{C}_{s_{\ell}}$ is also an element of $\mathcal{B}$, and hence,
$I_{\ell}$ is an element of $\widehat{\mathcal{P}}_{\mathcal{B}}$. Moreover,
combining the fact that
$\max\mathcal{C}_{s_{\ell}}\in\\{\alpha\in\mathcal{C}_{s_{\ell}}\colon\alpha\geq\alpha_{2\ell}\\}$
with (3.1), we have that $\max\mathcal{C}_{s_{\ell}}$ is contained in
$I_{\ell}$. In conclusion,

$\mu_{\mathcal{B}}(I_{\ell-1},I_{\ell})=(\max\mathsf{C}(I_{\ell};I_{\ell-1}),(\alpha_{2\ell-1},\alpha_{2\ell}))=(\max\mathcal{C}_{s_{\ell}},(\alpha_{2\ell-1},\alpha_{2\ell})),$

where $I_{0}=I$. Since $s_{\ell}\leq s_{\ell+1}$, we have that
$\max\mathcal{C}_{s_{\ell}}\geq\max\mathcal{C}_{s_{\ell+1}}$, and, hence,
$\mu_{\mathcal{B}}(I_{\ell-1},I_{\ell})\succ\mu_{\mathcal{B}}(I_{\ell},I_{\ell+1})$.
We define the maximal chain $c\coloneqq(I_{0},I_{1},\ldots,I_{N})$ of the
interval $[I,J]$, and thus $\mu_{\mathcal{B}}(c)$ is decreasing.

Consider another maximal chain
$c^{\prime}=(I^{\prime}_{0},I^{\prime}_{1},\ldots,I^{\prime}_{N})$ of $[I,J]$.
Let $i$ be the smallest index such that $I_{i}\neq I^{\prime}_{i}$, and $j$
the smallest index such that $I_{i}\subset I^{\prime}_{j}$ with $j>i$. Since
one of $\alpha_{2i-1}$ and $\alpha_{2i}$ is contained in
$\mathsf{C}(I^{\prime}_{j};I^{\prime}_{j-1})$, we have that
$\mathsf{C}(I_{i};I_{i-1})\subset\mathsf{C}(I^{\prime}_{j};I^{\prime}_{j-1})\subset\mathcal{C}_{s_{i}}$.
Therefore,

$\max\mathcal{C}_{s_{i}}=\max\mathsf{C}(I_{i};I_{i-1})\leq\max\mathsf{C}(I^{\prime}_{j};I^{\prime}_{j-1})\leq\max\mathcal{C}_{s_{i}},$

and, hence,
$\max\mathsf{C}(I_{i};I_{i-1})=\max\mathsf{C}(I^{\prime}_{j};I^{\prime}_{j-1})=\max\mathcal{C}_{s_{i}}$.

Now, let us consider two cases where
$\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})\subset\mathcal{C}_{s_{i}}$ or
$\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})\not\subset\mathcal{C}_{s_{i}}$.

CASE 1.
$\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})\subset\mathcal{C}_{s_{i}}$ : In
this case,

$\max\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})\leq\max\mathcal{C}_{s_{i}}=\max\mathsf{C}(I^{\prime}_{j};I^{\prime}_{j-1}).$

If $\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})<\max\mathcal{C}_{s_{i}}$, we
have that
$\mu_{\mathcal{B}}(I^{\prime}_{i-1},I^{\prime}_{i})\prec\mu_{\mathcal{B}}(I^{\prime}_{j-1},I^{\prime}_{j})$.
Otherwise, i.e. ,
$\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})=\max\mathcal{C}_{s_{i}}$, we note
that $I^{\prime}_{i}\setminus I^{\prime}_{i-1}$ intersects with
$\mathcal{C}_{s_{i}}\setminus I_{i}$, and either $\alpha_{2i-1}$ or
$\alpha_{2i}$ is an element of $I^{\prime}_{j}\setminus I^{\prime}_{j-1}$.
From (3.1), we have $\mathcal{C}_{s_{i}}\setminus
I_{i}=\\{\alpha\in\mathcal{C}_{s_{i}}\colon\alpha<\alpha_{2i}\\}$, and then

$\min I^{\prime}_{i}\setminus I^{\prime}_{i-1}<\alpha_{2i}\leq\max
I^{\prime}_{j}\setminus I^{\prime}_{j-1},$

indicating that

$((\max I^{\prime}_{i}\setminus I^{\prime}_{i-1},\min I^{\prime}_{i}\setminus
I^{\prime}_{i-1}),(\max I^{\prime}_{j}\setminus I^{\prime}_{j-1},\min
I^{\prime}_{j}\setminus I^{\prime}_{j-1}))\notin\mathcal{R}_{1}.$

Moreover, since $I^{\prime}_{i}\setminus I^{\prime}_{i-1}\cap
I^{\prime}_{j}\setminus I^{\prime}_{j-1}=\emptyset$,

$((\max I^{\prime}_{i}\setminus I^{\prime}_{i-1},\min I^{\prime}_{i}\setminus
I^{\prime}_{i-1}),(\max I^{\prime}_{j}\setminus I^{\prime}_{j-1},\min
I^{\prime}_{j}\setminus I^{\prime}_{j-1}))\notin\mathcal{R}_{2}.$

Therefore, either they are not comparable or
$\mu_{\mathcal{B}}(I^{\prime}_{i-1},I^{\prime}_{i})\prec\mu_{\mathcal{B}}(I^{\prime}_{j-1},I^{\prime}_{j})$,
and, hence, $\mu_{\mathcal{B}}(c^{\prime})$ is not decreasing.

Furthermore, since $\alpha_{2i-1}$ and $\alpha_{2i}$ are the two largest
elements of $\mathcal{C}_{s_{i}}\setminus I_{i-1}$ and
$I_{i-1}=I^{\prime}_{i-1}$, it can be observed that

$\begin{cases}\alpha_{2i-1}=\max I^{\prime}_{i}\setminus
I^{\prime}_{i-1}>\alpha_{2i}>\min I^{\prime}_{i}\setminus
I^{\prime}_{i-1},&\mbox{if }\alpha_{2i-1}\in I^{\prime}_{i}\setminus
I^{\prime}_{i-1}\\\ \alpha_{2i-1}>\alpha_{2i}\geq\max I^{\prime}_{i}\setminus
I^{\prime}_{i-1}>\min I^{\prime}_{i}\setminus
I^{\prime}_{i-1},&\mbox{otherwise}.\end{cases}$

We conclude that
$\mu_{\mathcal{B}}(I_{i-1},I_{i})\succ\mu_{\mathcal{B}}(I^{\prime}_{i-1},I^{\prime}_{i})$,
signifying that $\mu_{\mathcal{B}}(c)$ lexicographically succeeds
$\mu_{\mathcal{B}}(c^{\prime})$.

CASE 2.
$\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})\not\subset\mathcal{C}_{s_{i}}$ :
Choose $1\leq t\leq r$ such that the set $\mathcal{C}_{t}$ includes the set
$\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1})$. Since

$\mathcal{C}_{0}\cup\cdots\cup\mathcal{C}_{s-1}\subset
I_{i-1}=I^{\prime}_{i-1},$

we have $s<t$, and then

$\max\mathsf{C}(I_{i};I_{i-1})=\max\mathsf{C}(I^{\prime}_{j};I^{\prime}_{j-1})=\max\mathcal{C}_{s}>\max\mathcal{C}_{t}\geq\mathsf{C}(I^{\prime}_{i};I^{\prime}_{i-1}).$

Therefore, both $\mu_{\mathcal{B}}(I_{i-1},I_{i})$ and
$\mu_{\mathcal{B}}(I^{\prime}_{j-1},I_{j})$ succeed
$\mu_{\mathcal{B}}(I^{\prime}_{i-1},I^{\prime}_{i})$. Hence,
$\mu_{\mathcal{B}}(c^{\prime})$ is not decreasing, and $\mu_{\mathcal{B}}(c)$
lexicographically succeeds $\mu_{\mathcal{B}}(c^{\prime})$.

In conclusion, by CASES 1 and 2, $\mu_{\mathcal{B}}$ is reversed edge
lexicographical. ∎

A simplicial complex is _pure_ if all its maximal simplices has the same
dimension. A finite pure $d$-dimensional simplicial complex is said to be
_shellable_ if there exists an ordering $F_{1},F_{2},\ldots$ of its maximal
simplices, say a _shelling_ , such that $(\bigcup_{i=1}^{k-1}F_{i})\cap F_{k}$
is pure and of dimension $d-1$ for each $k\geq 2$.

Let
$\mathcal{P}_{\mathcal{B}}=\widehat{\mathcal{P}}_{\mathcal{B}}\setminus\\{\emptyset,I\\}$
be a partially ordered set whose order is given by inclusion. Given that $F$
is a facet of the order complex $\Delta(\mathcal{P}_{\mathcal{B}})$ if and
only if $F\cup\\{\emptyset,I\\}$ is a facet of
$\Delta(\widehat{\mathcal{P}}_{\mathcal{B}})$.

######  Theorem 3.5.

Let $\mathcal{B}$ be a chordal building set on a finite set of cardinality
$2k$. Then, $\Delta(\mathcal{P}_{\mathcal{B}})$ is shellable, and hence, its
geometric realization is homotopy equivalent to a bouquet of
$(k-2)$-dimensional spheres; that is,

$\Delta(\mathcal{P}_{\mathcal{B}})\simeq\bigvee^{\mathsf{a}(\mathcal{B})}S^{k-2},$

where $\mathsf{a}(\mathcal{B})$ denotes the number of spheres.

###### Proof.

From [1] and [2], if a bounded graded poset $\mathcal{P}$ is EL-shellable, the
order complex $\Delta(\mathcal{P})$ of $\mathcal{P}$ is shellable. By Lemma
3.4, the order complex $\Delta(\widehat{\mathcal{P}}_{\mathcal{B}})$ is
shellable, and so is $\Delta(\mathcal{P}_{\mathcal{B}})$. In addition, it is
well known that every shellable complex is Cohen-Macaulay [39], and then, its
geometric realization is homotopy equivalent to a bouquet of spheres of the
same dimension. Since the length of maximal chains of
$\mathcal{P}_{\mathcal{B}}$ is $k-1$, the dimension of each sphere is $k-2$,
as desired.

∎

##  4\. Alternating $\mathcal{B}$-permutations

Let $S$ be a finite set with even cardinality. This section is devoted to
explicitly computing the value of $\mathsf{a}(\mathcal{B})$ defined in Theorem
3.5 for a chordal building set $\mathcal{B}$ on $S$. A (one-line notation)
permutation $x=(x_{1}x_{2}\cdots x_{k})$ on $S$ is called a _$\mathcal{B}$
-permutation_ if for each $1\leq i\leq k$, $x_{i}$ and
$\max{\\{x_{1},x_{2},\ldots,x_{i}\\}}$ lie in the same connected component of
the restricted building set $\mathcal{B}|_{\\{x_{1},x_{2},\ldots,x_{i}\\}}$. A
$\mathcal{B}$-permutation $(x_{1}x_{2}\ldots x_{k})$ is called an _alternating
$\mathcal{B}$-permutation_ if $x_{1}>x_{2}<x_{3}>\cdots$. Inspired by the
assumption that the cardinality of $S$ is even, we utilize the one-line
notation $(x_{1}x_{2}/\cdots/x_{2k-1}x_{2k})$ to represent a permutation on
$S$, which is grouped into blocks of length two, separated by a slash.

######  Example 4.1.

We remark that the number of alternating $\mathcal{B}$-permutations is not an
invariant of nestohedra. Let us consider two labeling on path graphs with $4$
vertices in Example 2.3. Although the their corresponding nestohedra are
congruent as polytopes in $\mathbb{R}^{4}$, but the number of alternating
$\mathcal{B}$-permutations are different.

  1. (1)

Consider the labeling on $P_{4}$ as

$2$$3$$1$$4$

.

With this labeling, the alternating $\mathcal{B}(P_{4})$-permutations are
$(31/42)$, $(32/41)$ and $(41/32)$.

  2. (2)

Consider a path graph $P_{4}$ of order $4$ labeled as

$1$$2$$3$$4$

.

There are two alternating $\mathcal{B}(P_{4})$-permutations $(21/43)$ and
$(32/41)$.

One can immediately obtain the following proposition.

######  Proposition 4.2.

If a building set $\mathcal{B}$ on $S$ has an odd order connected component,
then there is no alternating $\mathcal{B}$-permutation.

###### Proof.

For a $\mathcal{B}$-permutation $x=(x_{1}x_{2}/\cdots/x_{2k-1}x_{2k})$ on $S$,
there is $1\leq i\leq k$ such that $x_{2i-1}$ and $x_{2i}$ are not contained
in the same connected component of $\mathcal{B}|_{\\{x_{1},\ldots,x_{2i}\\}}$.
In other words, $\max\\{x_{1},\ldots,x_{2i-1}\\}$ and
$\max\\{x_{1},\ldots,x_{2i}\\}$ are different, and then $x_{2i-1}<x_{2i}$
which confirms that $x$ is not alternating. ∎

Let $\mathcal{B}$ be a chordal building set on $S$, and
$c=(I_{0},I_{1},\ldots,I_{k})$ a maximal chain of
$\widehat{\mathcal{P}}_{\mathcal{B}}$. For each $1\leq i<k$, we say that
$\mu_{\mathcal{B}}(c)$ has a _decreasing position at $i$_ if

$\mu_{\mathcal{B}}(I_{i-1},I_{i})\succ\mu_{\mathcal{B}}(I_{i},I_{i+1}).$

######  Lemma 4.3.

Let $\mathcal{B}$ be a chordal building set on $S$ with no connected component
of odd order. There is a one-to-one correspondence between the set of
alternating $\mathcal{B}$-permutations and the set of maximal chains of
$\widehat{\mathcal{P}}_{\mathcal{B}}$ with no decreasing position.

###### Proof.

Let $\mathcal{F}$ denote the set of alternating $\mathcal{B}$-permutations,
and $\mathcal{G}$ the set of maximal chains $c$ of
$\widehat{\mathcal{P}}_{\mathcal{B}}$ such that $\mu_{\mathcal{B}}(c)$ has no
decreasing position.

Consider an alternating $\mathcal{B}$-permutation
$x=(x_{1}x_{2}/\cdots/x_{2k-1}x_{2k})$. For each $1\leq i\leq k$, since
$x_{2i-1}>x_{2i}$, it follows that
$\max\\{x_{1},\ldots,x_{2i-1}\\}=\max\\{x_{1},\ldots,x_{2i}\\}$, and then
$x_{2i-1}$ and $x_{2i}$ are contained in the same connected component of
$\mathcal{B}$. For each $1\leq i\leq k$, we have

$I^{x}_{i}\coloneqq\\{x_{1},\ldots,x_{2i}\\}\in\widehat{\mathcal{P}}_{\mathcal{B}}.$

For each $1\leq i<k$, considering the fact that
$\max\\{x_{1},\ldots,x_{2i+1}\\}$ and $x_{2i+1}$ belong to the same connected
component of $\mathcal{B}|_{\\{x_{1},\ldots,x_{2i+1}\\}}$, we have

$\max\mathsf{C}(I^{x}_{i};I^{x}_{i-1})\leq\max\\{x_{1},\ldots,x_{2i+1}\\}=\max\mathsf{C}(I^{x}_{i+1};I^{x}_{i}).$

Since $x_{2i}<x_{2i+1}$ and $x_{2i-1}\neq x_{2i+1}$, the pair
$((x_{2i-1},x_{2i}),(x_{2i+1},x_{2i+2}))$ is neither an element of
$\mathcal{R}_{1}$ nor $\mathcal{R}_{2}$, which confirms that

$\mu_{\mathcal{B}}(I_{i-1},I_{i})\not\succ\mu_{\mathcal{B}}(I_{i},I_{i+1}).$

Hence, for each alternating $\mathcal{B}$-permutation $x$, the maximal chain

$\Phi(x)\coloneqq(\emptyset=I^{x}_{0},I^{x}_{1},\ldots,I^{x}_{k})$

has no decreasing position, and then $\Phi$ can be regarded as an injective
map from $\mathcal{F}$ to $\mathcal{G}$.

Let $c=(I_{0},I_{1},\ldots,I_{k})\in\mathcal{G}$. For each $1\leq i\leq k$, we
define the pair $(x^{c}_{2i-1},x^{c}_{2i})$ of $S$ as

$(x^{c}_{2i-1},x^{c}_{2i})=(\max I_{i}\setminus I_{i-1},\min I_{i}\setminus
I_{i-1}).$

Define a permutation
$\Psi(c)\coloneqq(x^{c}_{1}x^{c}_{2}/\cdots/x^{c}_{2k-1}x^{c}_{2k})$ on $S$.
We will prove the following statement by mathematical induction on $k$:

(4.1) $\Psi(c)$ is an alternating $\mathcal{B}$-permutation.

To begin with, it is obviously confirmed that $\Psi(c)=(x^{c}_{1}x^{c}_{2})$
is an alternating $\mathcal{B}$-permutation. Let us assume that (4.1) holds
for $k=N-1$, where $N\geq 2$. Let $c=(I_{0},\ldots,I_{N})$ be a maximal chain
of $\widehat{\mathcal{P}}_{\mathcal{B}}$ such that $\mu_{\mathcal{B}}(c)$ has
no decreasing position. Consider a restricted building set
$\mathcal{B}^{\prime}\coloneqq\mathcal{B}|_{I_{1}\cup\cdots\cup I_{N-1}}$ of
$\mathcal{B}$. Note that $c^{\prime}=(I_{0},\ldots,I_{N-1})$ is a maximal
chain of $\widehat{\mathcal{P}}_{\mathcal{B}^{\prime}}$ and $c^{\prime}$ also
has no decreasing position. By the assumption, the permutation
$\Psi(c^{\prime})$ is an alternating $\mathcal{B}^{\prime}$-permutation. To
show that $\Psi(c)=(x^{c}_{1}x^{c}_{2}/\cdots/x^{c}_{2N-1}x^{c}_{2N})$ is an
alternating $\mathcal{B}$-permutation, it suffices to demonstrate that the
following two conditions hold:

  1. (C1)

$x^{c}_{2N-2}<x^{c}_{2N-1}$, and

  2. (C2)

$x^{c}_{2N-1}$ and $\max\\{x^{c}_{1},\ldots,x^{c}_{2N-1}\\}$ lie in the same
connected component of $\mathcal{B}|_{\\{x^{c}_{1},\ldots,x^{c}_{2N-1}\\}}$.

Put $x^{c}_{\ell}\coloneqq\max\\{x^{c}_{1},\ldots,x^{c}_{2N-2}\\}$, where
$1\leq\ell\leq 2N-2$. By the assumption, since $\Psi(c^{\prime})$ is a
$\mathcal{B}$-permutation, $x^{c}_{\ell}$ and $x^{c}_{2N-2}$ lie in the same
connected component of $\mathcal{B}^{\prime}$. It follows that $x^{c}_{\ell}$
is an element of $\mathsf{C}(I_{N-1};I_{N-2})$, and then,
$x^{c}_{\ell}=\max\mathsf{C}(I_{N-1};I_{N-2})$. We divide our proof into two
distinct cases;
$\max\mathsf{C}(I_{N-1};I_{N-2})<\max\mathsf{C}(I_{N};I_{N-1})$, or not.

Consider the first case. We have

$x^{c}_{\ell}=\max\mathsf{C}(I_{N-1};I_{N-2})<\max\mathsf{C}(I_{N};I_{N-1}).$

This inequality implies that $\max\mathsf{C}(I_{N};I_{N-1})$ is either
$x^{c}_{2N-1}$ or $x^{c}_{2N}$. Considering $x^{c}_{2N-1}>x^{c}_{2N}$, we
obtain $\max\mathsf{C}(I_{N};I_{N-1})=x^{c}_{2N-1}$. In conclusion, the
conditions (C1) and (C2) hold for $c$, which confirms that $\Psi(c)$ is an
alternating $\mathcal{B}$-permutation.

Now, we consider the case where
$\max\mathsf{C}(I_{N-1};I_{N-2})\geq\max\mathsf{C}(I_{N};I_{N-1})$. Note that
$c$ does not have a decreasing position at $(N-1)$, that is,

(4.2)
$\mu_{\mathcal{B}}(I_{N-2},I_{N-1})\not\succ\mu_{\mathcal{B}}(I_{N-1},I_{N}).$

It can be observed that
$\mathsf{C}(I_{N-1};I_{N-2})=\mathsf{C}(I_{N};I_{N-1})$ and
$x^{c}_{\ell}>x^{c}_{2N-1}>x^{c}_{2N}$. By the chordality of the building set
$\mathcal{B}$, we have that

$\mathcal{C}\coloneqq\\{x\in\mathsf{C}(I_{N};I_{N-1})\colon x>x^{c}_{2N}\\}$

is also an element of $\mathcal{B}$. Thus,

$\\{x^{c}_{\ell},x^{c}_{2N-1}\\}\subset\mathcal{C}\in\mathcal{B}|_{\\{x^{c}_{1},\ldots,x^{c}_{2N-1}\\}}.$

Since $x^{c}_{\ell}$ is also $\max\\{x^{c}_{1},\ldots,x^{c}_{2N-1}\\}$, the
condition (C2) holds for $c$. By (4.2),

$((x^{c}_{2N-3},x^{c}_{2N-2}),(x^{c}_{2N-1},x^{c}_{2N}))\notin\mathcal{R}_{1},$

and hence, the condition (C1) holds for $c$.

By mathematical induction, $\Psi(c)$ is an alternating
$\mathcal{B}$-permutation for all $c\in\mathcal{G}$. Since $\Phi(\Psi(c))=c$
for $c\in\mathcal{G}$, $\Phi$ is surjective, as desired. ∎

######  Theorem 4.4.

Let $\mathcal{B}$ be a chordal building set. Then, $\mathsf{a}(\mathcal{B})$
is the number of alternating $\mathcal{B}$-permutations.

###### Proof.

Let $\mathcal{B}$ be a chordal building set on $S$ of size $2k$. There is an
one-to-one correspondence between $(\ell-1)$-simplices in
$\Delta(\mathcal{P}_{\mathcal{B}})$ and chains of length $\ell$ in
$\mathcal{P}_{\mathcal{B}}$ for $1\leq\ell\leq k-1$. Then each
$(\ell-1)$-simplex can be regarded as an $\ell$-length chain of
$\mathcal{P}_{\mathcal{B}}$.

Let an $\ell$-length chain $c=(S_{1},S_{2},\ldots,S_{\ell})$ of
$\mathcal{P}_{\mathcal{B}}$ be given, where $1\leq\ell\leq k$. For
convenience, let $S_{0}\coloneqq\emptyset$ and $S_{\ell+1}\coloneqq S$. For
each $1\leq i\leq\ell+1$, according to Lemma 3.4, there is the unique maximal
chain

$c^{i}=(S_{i-1}=I^{c}_{k_{0}+\cdots+k_{i-1}},I^{c}_{k_{0}+\cdots+k_{i-1}+1},\ldots,I^{c}_{k_{0}+\cdots+k_{i-1}+k_{i}}=S_{i}),$

of the interval $[S_{i-1},S_{i}]$ of $\widehat{\mathcal{P}}_{\mathcal{B}}$
such that $\mu_{\mathcal{B}}(c^{i})$ is decreasing, where $k_{0}=0$, and
$2k_{i}=\left|S_{i}\setminus S_{i-1}\right|$. In conclusion, for each
$\ell$-length chain $c$ of $\mathcal{P}_{\mathcal{B}}$, there is the induced
maximal chain

$\bar{c}\coloneqq(I^{c}_{0},I^{c}_{1},\ldots,I^{c}_{k})$

of $\widehat{\mathcal{P}}_{\mathcal{B}}$.

Let us consider a maximal chain $d=(I_{0},I_{1},\ldots,I_{k})$ of
$\widehat{\mathcal{P}}_{\mathcal{B}}$ with $N$ decreasing positions, where
$1\leq N\leq k-1$. Define the set

$\mathcal{I}_{d}=\\{1\leq i\leq k-1\colon\text{ $d$ does not have a decreasing
position at $i$.}\\}$

of indices. Let $c=(I_{i_{1}},\ldots,I_{i_{\ell}})$ be an $\ell$-length chain
of $\mathcal{P}_{\mathcal{B}}$. The induced maximal chain $\bar{c}$ of
$\widehat{\mathcal{P}}_{\mathcal{B}}$ is equal to $d$ if and only if

$\mathcal{I}_{d}\subset\\{i_{1},i_{2},\ldots,i_{\ell}\\}.$

Consequently, the number of $\ell$-length chains of
$\mathcal{P}_{\mathcal{B}}$ that induces $d$ is given by

${N\choose\ell-\left|\mathcal{I}_{d}\right|}={N\choose\ell-(k-1-N)}={N\choose
k-1-\ell},$

if $k-1-N\leq\ell\leq k-1$, and $0$ otherwise.

Let $\mathsf{a}_{N}$ denote the number of maximal chains of
$\widehat{\mathcal{P}}_{\mathcal{B}}$ with $N$ decreasing positions for each
$0\leq N\leq k-1$. The total number of $\ell$-length chains of
$\mathcal{P}_{\mathcal{B}}$ is

$\sum_{N=k-1-\ell}^{k-1}{N\choose k-1-\ell}\mathsf{a}_{N}.$

Now, the Euler characteristic $\chi(\Delta(\mathcal{P}_{\mathcal{B}}))$ of
$\Delta(\mathcal{P}_{\mathcal{B}})$ is

$\displaystyle\chi(\Delta(\mathcal{P}_{\mathcal{B}}))$
$\displaystyle=\sum_{\ell=1}^{k-1}{(-1)^{\ell-1}\\#\ell\text{-length chains of
}}\mathcal{P}_{\mathcal{B}}$
$\displaystyle=\sum_{\ell=1}^{k-1}\sum_{N=k-1-\ell}^{k-1}(-1)^{\ell-1}{N\choose
k-1-\ell}\mathsf{a}_{N}$
$\displaystyle=\mathsf{a}_{k-1}+\sum_{\ell=0}^{k-1}\sum_{N=k-1-\ell}^{k-1}(-1)^{\ell-1}{N\choose
k-1-\ell}\mathsf{a}_{N}$
$\displaystyle=\mathsf{a}_{k-1}+\sum_{N=0}^{k-1}\left(\sum_{\ell=k-1-N}^{k-1}(-1)^{\ell-1}{N\choose
k-1-\ell}\right)\mathsf{a}_{N}.$

By the binomial theorem, we have that

$\sum_{\ell=k-1-N}^{k-1}(-1)^{\ell-1}{N\choose k-1-\ell}=0$

for all $1\leq N\leq k-1$, and, hence,
$\chi(\Delta(\mathcal{P}_{\mathcal{B}}))=\mathsf{a}_{k-1}+(-1)^{k-2}\mathsf{a}_{0}$.

We note that $\mathsf{a}_{k-1}=1$ by Lemma 3.4. In addition, by Theorem 3.5,
$\chi(\Delta(\mathcal{P}_{\mathcal{B}}))=1+(-1)^{k-2}\mathsf{a}(\mathcal{B})$.
Therefore, $\mathsf{a}(\mathcal{B})=\mathsf{a}_{0}$, that is the number of
alternating $\mathcal{B}$-permutations by Lemma 4.3.

∎

##  5\. Proof of the main theorem

In this section, we prove Theorem 1.1. For a connected building set
$\mathcal{B}$ on $[n+1]=\\{1,\ldots,n+1\\}$, the rational Betti number of
$X^{\mathbb{R}}_{\mathcal{B}}$ is determined by the reduced Betti numbers of
$(K_{\mathcal{B}})_{I}$ for even cardinality subset $I$ of $[n+1]$ by (2.2),
where $(K_{\mathcal{B}})_{I}$ the full subcomplex of $K_{\mathcal{B}}$ induced
by all vertices $J$, where $\left|J\cap I\right|$ is odd.

Let us consider the case when $n+1$ is even, and $I=[n+1]$. We denote by
$K_{\mathcal{B}}^{\operatorname{odd}}$ (respectively,
$K_{\mathcal{B}}^{\operatorname{even}}$) the induced subcomplex of a nested
set complex $K_{\mathcal{B}}$ whose vertices have odd (respectively, even)
cardinality. In other words,
$K_{\mathcal{B}}^{\operatorname{odd}}=(K_{\mathcal{B}})_{[n+1]}$ and
$K_{\mathcal{B}}^{\operatorname{even}}$ is the complement of the simplicial
complex of $K_{\mathcal{B}}^{\operatorname{odd}}$ in $K_{\mathcal{B}}$. Since
the geometric realization of $K_{\mathcal{B}}$ is topologically homeomorphic
to the $(2k-2)$-dimensional sphere $S^{2k-2}$, by the Alexander duality,
$H_{\ast}(K_{\mathcal{B}}^{\operatorname{odd}})$ is completely determined by
$H^{\ast}(K_{\mathcal{B}}^{\operatorname{even}})$. In addition, with the
coefficients of (co)homology being in a field $\mathbb{Q}$, by the Universal
Coefficient Theorem, we have

(5.1)
$\tilde{H}_{i}(K_{\mathcal{B}}^{\operatorname{odd}})\cong\tilde{H}_{2k-3-i}(K_{\mathcal{B}}^{\operatorname{even}})\text{
for all $i\geq 0$}.$

Refer [31, Theorem 71.1 and 53.5] for the Alexander duality and the Universal
Coefficient Theorem, respectively.

######  Lemma 5.1.

The geometric realizations of $\Delta(\mathcal{P}_{\mathcal{B}})$ and
$K_{\mathcal{B}}^{\operatorname{even}}$ are homeomorphic. In addition, if
$\mathcal{B}$ is chordal, the $i$th reduced rational Betti numbers
$\tilde{\beta}_{i}$ of $K_{\mathcal{B}}^{\operatorname{odd}}$ are

$\tilde{\beta}_{i}(K_{\mathcal{B}}^{\operatorname{odd}})=\begin{cases}\mathsf{a}(\mathcal{B}),&\mbox{if
}i=k-1,\\\ 0,&\mbox{otherwise}.\end{cases}$

###### Proof.

A proof of the former statement of this lemma, limited to graphical building
sets, is provided in [11, Lemma 4.7]. However, upon closer inspection, the
proof does not actually rely on the condition of being a graphical building
set, confirming that the lemma holds for any building sets. The latter
statement follows from the preceding statement, Theorem 3.5 and (5.1). ∎

Now, let us consider more general cases. For any simplicial complex $K$, the
_link_ $\operatorname{Lk}_{K}(v)$ of a vertex $v$ in $K$ is the set of all
faces $\sigma\in K$ such that $v\notin\sigma$ and $\sigma\cup\\{v\\}\in K$. It
is well known that $\operatorname{Lk}_{K}(v)$ is a full subcomplex of $K$. If
$\operatorname{Lk}_{K}(v)$ is a topological cone, then $K$ is homotopy
equivalent to the simplicial complex
$K-v\coloneqq\\{\sigma\setminus\\{v\\}\colon\sigma\in K\\}$.

A simplicial complex $K$ is said to be _flag_ if it contains every clique of
its $1$-skeleton. It can be immediately observed that an induced subcomplex of
a flag complex is also a flag complex. For each subset $I$ of $[n+1]$ with
even cardinality, we shall discuss an induced subcomplex
$(K_{\mathcal{B}})_{I}$ in order to compute the Betti numbers of
$X_{\mathcal{B}}^{\mathbb{R}}$. If a nested set complex $K_{\mathcal{B}}$ is
flag, then all induced subcomplexes $(K_{\mathcal{B}})_{I}$ are also a flag
complex. For each vertex $X$ in $(K_{\mathcal{B}})_{I}$, a vertex $Y$ in
$(K_{\mathcal{B}})_{I}$ is also a vertex in
$\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X)$ if and only if the pair $(X,Y)$
holds the following conditions:

(5.2) $X\subsetneq Y,Y\subsetneq X,\text{ or }X\cup Y\notin\mathcal{B}.$

######  Lemma 5.2.

Assume that a nested set complex $K_{\mathcal{B}}$ is flag. For any
$I\subset[n+1]$ with even cardinality, an induced subcomplex
$(K_{\mathcal{B}})_{I}$ is homotopy equivalent to
$K_{\mathcal{B}|_{I}}^{\text{odd}}$.

###### Proof.

Define the partial order $\trianglelefteq$ of the set of vertices of
$(K_{\mathcal{B}})_{I}$ as $X_{1}\trianglelefteq X_{2}$ if $X_{1}\cap
I\subseteq X_{2}\cap I$ for each pair $(X_{1},X_{2})$ of vertices in
$(K_{\mathcal{B}})_{I}$. Let $\mathcal{V}$ be the set of vertices in
$(K_{\mathcal{B}})_{I}$ that are not in $I$, with the understanding that
$\mathcal{V}$ is nonempty, since the case where $\mathcal{V}=\emptyset$ is
straightforward. For each $1\leq i\leq N$, we define $\mathcal{V}_{i}$ as the
set of minimal elements of

$\mathcal{U}_{i}=\mathcal{V}\setminus(\bigcup_{0\leq j\leq
i-1}\mathcal{V}_{j})$

with respect to the order $\trianglelefteq$, where $\mathcal{V}_{0}=\emptyset$
and $N$ is a sufficiently large positive integer.

Now, for each $1\leq i\leq N$, let $(K_{\mathcal{B}})_{I}^{i}$ denote the
simplicial complex obtained by removing from $(K_{\mathcal{B}})_{I}$ belonging
to $\bigcup_{0\leq j\leq i-1}\mathcal{V}_{j}$. Let $X\in\mathcal{V}_{i}$ be a
given. Considering that $\left|X\cap I\right|$ is odd, one can choose a
connected component $X_{0}$ of $\mathcal{B}|_{X\cap I}$ with odd cardinality.
For each vertex $Y$ of $\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X)$, the
pair $(X,Y)$ satisfies (5.2). To show that either $Y=X_{0}$ or
$Y\in\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X_{0})$, we consider (5.2) by
dividing it into three cases.

  1. (1)

If $X\subsetneq Y$, then $X_{0}\subseteq X\subsetneq Y$ which confirms that
$Y\in\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X)$.

  2. (2)

Let $Y\subsetneq X$. If $Y\subseteq I$, since $Y\in\mathcal{B}|_{X\cap I}$ and
$X_{0}$ is a connected component of $\mathcal{B}|_{X\cap I}$, either
$Y\subseteq X_{0}$ or $Y\cup X_{0}\notin\mathcal{B}$. If $Y\not\subset I$,
then $Y\in\mathcal{V}_{j}$ for $j\geq i$. Since $Y\cap I\subseteq X\cap I$ and
$X\subset\mathcal{V}_{i}$, we have $Y\cap I=X\cap I$, and then

$X_{0}\subseteq X\cap I=Y\cap I\subseteq Y.$

Thus, $Y=X_{0}$ or $Y\in\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X_{0})$.

  3. (3)

Let $X\cup Y\notin\mathcal{B}$. If $X_{0}\cup Y\in\mathcal{B}$, since
$X_{0}\subset X$, then

$X\cup Y=X\cup(X_{0}\cup Y)\in\mathcal{B},$

which contradicts that $X\cup Y\notin\mathcal{B}$. Hence, $X_{0}\cup Y$ is not
an element of $\mathcal{B}$. It follows that
$Y\in\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X)$.

In conclusion, the link $\operatorname{Lk}_{(K_{\mathcal{B}})_{I}}(X)$ of $X$
in $(K_{\mathcal{B}})_{I}^{i}$ is a topological cone whose apex is $X_{0}$.
Hence, one can remove each elements of $\mathcal{V}_{i}$ from
$(K_{\mathcal{B}})_{I}^{i}$, ensuring that its homotopy type remains
unchanged.

Therefore, from $\mathcal{V}_{1}$ to $\mathcal{V}_{N}$, one can sequentially
remove all elements in $\mathcal{V}$ from $(K_{\mathcal{B}})_{I}$ without
changing its homotopy type. Since $(K_{\mathcal{B}})_{I}$ with all elements of
$\mathcal{V}$ removed is isomorphic to $K_{\mathcal{B}|_{I}}^{\text{odd}}$, we
finish the proof of the lemma. ∎

######  Lemma 5.3.

Let $K_{\mathcal{B}}$ be a flag nested set complex. For each subset
$I\subset[n+1]$ with even cardinality, if $\mathcal{B}|_{I}$ has a connected
component of odd order, then $K_{\mathcal{B}|_{I}}^{\text{odd}}$ is
contractible.

###### Proof.

Choose a connected component $I_{0}$ of $\mathcal{B}|_{I}$ with odd
cardinality. Let $Y$ be a vertex of $K_{\mathcal{B}|_{I}}^{\text{odd}}$. If
$Y\subseteq I_{0}$, then either $Y=I_{0}$ or $(I_{0},Y)$ satisfies (5.2).

Thus, we assume that $Y\not\subset I_{0}$. Since $I_{0}$ is a component of
$\mathcal{B}|_{I}$, we have $Y\cup I_{0}\notin\mathcal{B}|_{I}$. Hence,
$(I_{0},Y)$ holds (5.2), which confirms that
$K_{\mathcal{B}|_{I}}^{\text{odd}}$ is a topological cone with its apex is
$I_{0}$, as desired. ∎

Now, we introduce the main result, restating Theorem 1.1.

######  Theorem 5.4.

For a connected chordal building set $\mathcal{B}$ on
$[n+1]=\\{1,2,\ldots,n+1\\}$, the $k$th Betti number of
$X^{\mathbb{R}}_{\mathcal{B}}$ is

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}})=\sum_{I\in{[n+1]\choose
2k}}\mathsf{a}(\mathcal{B}|_{I}),$

where $\mathsf{a}(\mathcal{B}|_{I})$ is the number of alternating
$\mathcal{B}|_{I}$-permutations.

###### Proof.

From [36, Proposition 9.7], if a building set $\mathcal{B}$ is chordal, then a
nested set complex $K_{\mathcal{B}}$ is a flag complex. Combining (2.2) with
Lemma 5.2, we have

$\beta_{k}(X_{\mathcal{B}}^{\mathbb{R}})=\sum_{\begin{subarray}{c}I\subset[n+1]\\\
\left|I\right|\text{ is
even}\end{subarray}}\widetilde{\beta}_{k-1}(K^{\operatorname{odd}}_{\mathcal{B}|_{I}}).$

Since every restricted building set of a chordal building set is chordal, for
a subset $I$ of $\widehat{\mathcal{P}}_{\mathcal{B}}$ with
$\left|I\right|=2k$, we have
$\tilde{\beta}_{i}(K_{\mathcal{B}|_{I}}^{\text{odd}})=\mathsf{a}(\mathcal{B}|_{I})\cdot\delta_{i,k-1}$
for all integers $i$, where $\delta_{i,j}$ is the Kronecker delta by Lemma
5.1. We have

$\sum_{I\in\hat{\mathcal{P}}_{\mathcal{B}}}\widetilde{\beta}_{k-1}(K_{\mathcal{B}})_{I}=\sum_{\begin{subarray}{c}I\in\hat{\mathcal{P}}_{\mathcal{B}}\\\
\left|I\right|=2k\end{subarray}}\mathsf{a}(\mathcal{B}|_{I}).$

Let $I$ be a subset of $[n+1]$ such that
$I\notin\widehat{\mathcal{P}}_{\mathcal{B}}$. By Proposition 4.2 and Lemma
5.3, $\mathsf{a}(\mathcal{B}|_{I})$ and
$\tilde{\beta}_{i}(K^{\operatorname{odd}}_{\mathcal{B}|_{I}})$ both become
equal to $0$ for all $i$, respectively. Consequently, one can write that

$\sum_{\begin{subarray}{c}I\notin\hat{\mathcal{P}}_{\mathcal{B}}\\\
\left|I\right|\text{ is
even}\end{subarray}}(K_{\mathcal{B}})_{I}=0=\sum_{\begin{subarray}{c}I\notin\hat{\mathcal{P}}_{\mathcal{B}}\\\
\left|I\right|=2k\end{subarray}}\mathsf{a}(\mathcal{B}|_{I}).$

Therefore,

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}})=\sum_{\begin{subarray}{c}I\subset[n+1]\\\
\left|I\right|\text{ is
even}\end{subarray}}\widetilde{\beta}_{k-1}(K^{\operatorname{odd}}_{\mathcal{B}|_{I}})=\sum_{\left|I\right|=2k}\mathsf{a}(\mathcal{B}|_{I})$

for each integers $k$, where $\mathsf{a}(\mathcal{B}|_{I})$ is the number of
alternating $\mathcal{B}|_{I}$-permutations by Theorem 4.4. ∎

######  Remark 5.5.

Let $G$ be a finite simple graph, and let $I$ be a subset of the vertex set
$V(G)$ of $G$ with even cardinality $2k$. Note that the $k$th Betti number
$\beta_{k}((K_{\mathcal{B}(G)})_{I})$ of $(K_{\mathcal{B}(G)})_{I}$ is the
$a$-number $a(G|_{I})$ of $G|_{I}$. If $G$ is a chordal graph with vertices
labeled by a perfect elimination ordering, the number
$\mathsf{a}(\mathcal{B}(G)|_{I})$ of alternating
$\mathcal{B}(G)|_{I}$-permutation is exactly
$\beta_{k}((K_{\mathcal{B}(G)})_{I})$, which confirms
$\mathsf{a}(\mathcal{B}(G)|_{I})$ is equal to the $a$-number $a(G|_{I})$.

##  6\. Examples

Let us consider several examples which illustrate Theorem 1.1.

###  6.1. Permutohedra

For a complete graph $K_{n+1}$ with vertices $1,2,\ldots,n+1$, the graphical
building set $\mathcal{B}=\mathcal{B}(K_{n+1})$ consists of all nonempty
subsets of $[n+1]=\\{1,\ldots,n+1\\}$. Every permutation on $[n+1]$ is also a
$\mathcal{B}$-permutation, which proves (1) of Corollary 1.2.

###  6.2. Associahedra

Let $P_{2k}$ denote a path graph having $2k$ vertices labeled consecutively as
$1,2,\ldots,2k$. One can verify that a permutation $x=(x_{1}x_{2}\cdots
x_{2k})$ is a $\mathcal{B}(P_{2k})$-permutation if and only if $x$ is _$312$
-avoiding_ i.e. there are no $1\leq i<j<l\leq 2k$ such that
$x_{j}<x_{l}<x_{i}$. See [36, Section 10.2] for more details.

###### Proof for (2) of Corollary 1.2.

Consider a path graph $G=P_{n+1}$ with $n+1$ vertices labeled consecutively as
$1,2,\ldots,n+1$, and a subset $I\subset[n+1]$ whose cardinality is $2k$. If
$G|_{I}$ has a connected component of odd order, then there is no alternating
$\mathcal{B}(G)|_{I}$-permutation by Proposition 4.2.

Assume that $G|_{I}$ has no component of odd order. Then $G|_{I}$ consists of
path graphs $G_{I_{1}},\ldots,G_{I_{r}}$, each of which has an even order
$2k_{1},\ldots,2k_{r}$, respectively. We assume that $\max I_{i}+1<\min
I_{i+1}$ for all $1\leq i<r$. For a $\mathcal{B}(G)|_{I}$-permutation
$(x_{1}x_{2}\cdots x_{2k})$,

$I_{i}=\\{x_{2k_{0}+\cdots+2k_{i-1}+1},x_{2k_{0}+\cdots+2k_{i-1}+2},\ldots,x_{2k_{0}+\cdots+2k_{i-1}+2k_{i}}\\},$

where $k_{0}=0$. Consequently, we obtain that each
$\mathcal{B}(G)|_{I}$-permutation is given by a tuple of
$\mathcal{B}(G)|_{I_{i}}$-permutations for all $1\leq i\leq r$. Therefore, the
number of alternating $\mathcal{B}(G)|_{I}$-permutations is the product of the
numbers of $312$-avoiding alternating permutations on $I_{i}$ for all $1\leq
i\leq l$.

By applying Theorem 1.1, we conclude that

$\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}(P_{n+1})})=\sum_{I\in{[n+1]\choose
2k}}\prod_{1\leq i\leq r_{I}}\\#312\text{-avoiding alternating permutations on
}I_{i},$

where the induced subgraph $G|_{I}$ consists of the even order connected
components $G|_{I_{1}},\ldots,G|_{I_{r_{I}}}$. ∎

For instance, if we consider a path graph $P_{6}$ with six vertices labeled
consecutively as $1,\ldots,6$, then the $k$th Betti number $\beta_{k}$ of
$X^{\mathbb{R}}_{\mathcal{B}(P_{6})}$ is presented in Table 1.

| $I$ |  $312$-avoiding alternating permutations on $I$ | $\beta_{k}$
---|---|---|---
$k=0$ | $\emptyset$ | $\\{(\ )\\}$ | $1$
$k=1$ | $\\{1,2\\}$ | $\\{(21)\\}$ |
| $\\{2,3\\}$ | $\\{(32)\\}$ |
| {3,4} | $\\{(43)\\}$ | $5$
| {4,5} | $\\{(54)\\}$ |
| {5,6} | $\\{(65)\\}$ |
$k=2$ | $\\{1,2,3,4\\}$ | $\\{(2143),(3241)\\}$ |
| $\\{1,2,4,5\\}$ | $\\{(2154)\\}$ |
| $\\{1,2,5,6\\}$ | $\\{(2165)\\}$ | $9$
| $\\{2,3,4,5\\}$ | $\\{(3254),(4352)\\}$ |
| $\\{2,3,5,6\\}$ | $\\{(3265)\\}$ |
| $\\{3,4,5,6\\}$ | $\\{(4365),(5463)\\}$ |
$k=3$ | $\\{1,2,3,4,5,6\\}$ | $\\{(214365),(215463),(324165),(325461),(435261)\\}$ | $5$
Table 1. Nonzero Betti numbers of $X^{\mathbb{R}}_{\mathcal{B}(P_{6})}$

###  6.3. Stellohedra

Let $G=K_{1,n}$ denote the star graph with the central vertex $n+1$ connected
to the vertices $1,\ldots,n$. If $I\in{[n+1]\choose 2k}$ does not include
$n+1$ as a vertex, $\mathcal{B}(G)|_{I}$ has a component of odd order, and
then there is no alternating $\mathcal{B}(G)|_{I}$-permutation, as stated in
Proposition 4.2.

Now, consider the case where $I\in{[n]\choose 2k-1}$. Let $x=(x_{1}x_{2}\cdots
x_{2k})$ be an alternating permutation on $I\cup\\{n+1\\}$. Then $x$ forms a
$\mathcal{B}(G)|_{I}$-permutation if and only if $x_{1}=n+1$. In conclusion,
this implies (3) of Corollary 1.2.

###  6.4. Hochschild polytopes

In this section, we assume that $(m,n)$ is a pair of non-negative integers
such that $m+n$ is even. Denote by
$\mathrm{Alt}_{\mathds{H}\mathrm{och}(m,n)}$ the set of alternating
permutations $(x_{1}x_{2}\cdots x_{m+n})$ on $[m+n]=\\{1,\ldots,m+n\\}$ such
that for each $i,j\in[m+n]$;

$m+n\geq x_{i}>x_{j}\geq m+1$

implies that $i<j$.

Consider a building set $\mathcal{B}_{m,n}$ defined in Section 2.3.

######  Proposition 6.1.

For a permutation $x$ on $[m+n]$, $x$ is an alternating
$\mathcal{B}_{m,n}$-permutation if and only if
$x\in\mathrm{Alt}_{\mathds{H}\mathrm{och}(m,n)}$.

###### Proof.

Let $\mathcal{B}=\mathcal{B}_{m,n}$. If $x$ is an element of
$\mathrm{Alt}_{\mathds{H}\mathrm{och}(m,n)}$, it can be easily verified that
$x$ is an alternating $\mathcal{B}$-permutation. Then we only focus on the
part of ‘only if’.

Let $x=(x_{1}x_{2}\cdots x_{m+n})$ be an alternating
$\mathcal{B}$-permutation. Assume that $x$ is not an element of
$\mathrm{Alt}_{\mathds{H}\mathrm{och}(m,n)}$. Then one can choose a pair
$(i,j)$ of elements of $[m+n]$ such that

$i<j\text{ and }m+1\leq x_{i}<x_{j}\leq m+n.$

Note that the singleton $\\{x_{i}\\}$ is a connected component of the
restricted building set $\mathcal{B}|_{\\{x_{1},\ldots,x_{i}\\}}$. Since $x$
is a $\mathcal{B}$-permutation, we have that
$x_{i}=\max\\{x_{1},\ldots,x_{i}\\}$. If there exists $1\leq i^{\prime}<i$
such that $x_{i^{\prime}}>x_{i}$, it follows that

$\max\\{x_{1},\ldots,x_{i}\\}\geq x_{i^{\prime}}>x_{i}.$

which contradicts that $x_{i}=\max\\{x_{1},\ldots,x_{i}\\}$. Thus, we assume
that $\max\\{x_{1},\ldots,x_{i}\\}=x_{i}$. Then either $i=1$ or
$x_{i-1}<x_{i}$. Considering that $x$ is alternating, we have that
$x_{i}>x_{i+1}$. It follows that the singleton $\\{x_{i}\\}$ forms also a
connected component of $\mathcal{B}|_{\\{x_{1},\ldots,x_{i+1}\\}}$. We
conclude that $x_{i}$ and $x_{i+1}$ are not contained in the same connected
component of $\mathcal{B}|_{\\{x_{1},\ldots,x_{i+1}\\}}$, which contradicts
$x$ is a $\mathcal{B}$-permutation. ∎

For non-negative integers $m$ and $n$, by Proposition 6.1, the set
$\mathrm{Alt}_{\mathds{H}\mathrm{och}(m,n)}$ can be regarded as the set of
alternating $\mathcal{B}_{m,n}$-permutations.

######  Lemma 6.2.

Let $m$ and $n$ be non-negative integers such that $m+n$ is even. If $n>m+2$,
there is no alternating $\mathcal{B}_{m,n}$-permutation.

###### Proof.

Let us consider an alternating $\mathcal{B}_{m,n}$-permutation
$x=(x_{1}x_{2}\cdots x_{m+n})$. Note that, for each $1\leq
i\leq\frac{m+n}{2}$, $x_{2i}$ is the maximal element of
$\\{x_{2i},\ldots,x_{m+n}\\}$. Since $m+n$ is even, we have $n\geq m+4$. By
the pigeonhole principle, there exist $1\leq\ell<\frac{m+n}{2}$ such that

$\\{x_{2\ell-1},x_{2\ell}\\}\subset[m+1,m+n].$

Since $x_{2\ell}>x_{2\ell+1}$, it contradicts the fact that $x$ is
alternating. In conclusion, if $n\geq m+2$, there is no alternating
$\mathcal{B}_{m,n}$-permutation. ∎

Now, we prove Corollary 1.3.

###### Proof for Corollary 1.3.

Let an integer $0\leq k\leq\frac{m+n}{2}$ be given. For each $1\leq
r\leq\min\\{2k,n\\}$, let $\mathcal{I}_{r}$ be the set defined as

$\mathcal{I}_{r}=\left\\{I\in{[m+n]\choose 2k}\colon
I\cap[m+1,m+n]=[m+n-(r-1),m+n]\right\\},$

and $\mathcal{I}_{0}\coloneqq{[m]\choose 2k}$. We note that
$\left|\mathcal{I}_{r}\right|={m\choose 2k-r}$, and for each
$I\in\mathcal{I}_{r}$, the number of alternating
$\mathcal{B}_{m,n}|_{I}$-permutations is exactly
$\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(s,r)}\right|$, where $s=2k-r$.

For any given $I\in{[m+n]\choose 2k}$, by (2.3), the restricted building set
$\mathcal{B}_{m,n}|_{I}$ has no connected component of odd order if and only
if $I\cap[m+1,m+n]$ is either the empty set or $[m+r,m+n]$ for some $1\leq
r\leq n$. Combining Theorem 1.1 with Proposition 4.2, we conclude that

$\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)})=\sum_{s+r=2k}{m\choose
s}\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(s,r)}\right|,$

where $s$ and $r$ are non-negative integers.

Fix a positive integer $m$, and assume that $n\geq m+2$. From Lemma 6.2, the
cardinality of the set

$\\{(s,r)\colon s+r=2k,0\leq s\leq m,0\leq r\leq\min\\{n,s+2\\}\\}$

is constant. For each $n\geq m+2$, we conclude that

$\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)})=\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,m+2)})$

for all $k\geq 0$. ∎

######  Example 6.3.

Consider the building set $\mathcal{B}_{2,4}$. The $k$th Betti number
$\beta_{k}$ of $X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(2,4)}$ is computed as
in Table 2. In addition, the list of the Betti numbers
$\beta_{k}(X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)})$ of
$X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)}$ up to $m=8$ is given in Table
3. The Python codes used validation are available at

https://github.com/YounghanYoon/RealHochVar.

| $(s,r)$ | $\mathrm{Alt}_{\mathds{H}\mathrm{och}(s,r)}$ | $\beta_{k}$
---|---|---|---
$k=0$ | $(0,0)$ | $\\{(\ )\\}$ | $1$
$k=1$ | $(2,0)$ | $\\{(21)\\}$ |
| $(1,1)$ | $\\{(21)\\}$ | $\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(2,0)}\right|+2\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(1,1)}\right|+\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(0,2)}\right|=4$
| $(0,2)$ | $\\{(21)\\}$ |
$k=2$ | $(2,2)$ | $\\{(4231),(4132),(2143)\\}$ |
| $(1,3)$ | $\\{(4132)\\}$ | $\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(2,2)}\right|+2\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(1,3)}\right|+\left|\mathrm{Alt}_{\mathds{H}\mathrm{och}(0,4)}\right|=5$
| $(0,4)$ | $\emptyset$ |
$k=3$ | $(2,4)$ | $\\{(625143),(615243)\\}$ | $2$
Table 2. Nonzero Betti numbers of $X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(2,4)}$ $m$ | $n$ | $\beta_{0}$ | $\beta_{1}$ | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{5}$ | $\beta_{6}$ | $\beta_{7}$ | $\beta_{8}$ | $\beta_{9}$
---|---|---|---|---|---|---|---|---|---|---|---
$0$ | $\geq 2$ | $1$ | $1$ |  |  |  |  |  |  |  |
$1$ | $2$ | $1$ | $2$ |  |  |  |  |  |  |  |
| $\geq 3$ | $1$ | $2$ | $1$ |  |  |  |  |  |  |
$2$ | $2$ | $1$ | $4$ | $3$ |  |  |  |  |  |  |
| $3$ | $1$ | $4$ | $5$ |  |  |  |  |  |  |
| $\geq 4$ | $1$ | $4$ | $5$ | $2$ |  |  |  |  |  |
$3$ | $2$ | $1$ | $7$ | $14$ |  |  |  |  |  |  |
| $3$ | $1$ | $7$ | $17$ | $11$ |  |  |  |  |  |
| $4$ | $1$ | $7$ | $17$ | $17$ |  |  |  |  |  |
| $\geq 5$ | $1$ | $7$ | $17$ | $17$ | $6$ |  |  |  |  |
$4$ | $2$ | $1$ | $11$ | $43$ | $33$ |  |  |  |  |  |
| $3$ | $1$ | $11$ | $47$ | $77$ |  |  |  |  |  |
| $4$ | $1$ | $11$ | $47$ | $89$ | $52$ |  |  |  |  |
| $5$ | $1$ | $11$ | $47$ | $89$ | $76$ |  |  |  |  |
| $\geq 6$ | $1$ | $11$ | $47$ | $89$ | $76$ | $24$ |  |  |  |
$5$ | $2$ | $1$ | $16$ | $105$ | $226$ |  |  |  |  |  |
| $3$ | $1$ | $16$ | $110$ | $336$ | $241$ |  |  |  |  |
| $4$ | $1$ | $16$ | $110$ | $356$ | $501$ |  |  |  |  |
| $5$ | $1$ | $16$ | $110$ | $356$ | $561$ | $300$ |  |  |  |
| $6$ | $1$ | $16$ | $110$ | $356$ | $561$ | $420$ |  |  |  |
| $\geq 7$ | $1$ | $16$ | $110$ | $356$ | $561$ | $420$ | $120$ |  |  |
$6$ | $2$ | $1$ | $22$ | $220$ | $922$ | $723$ |  |  |  |  |
| $3$ | $1$ | $22$ | $226$ | $1142$ | $2169$ |  |  |  |  |
| $4$ | $1$ | $22$ | $226$ | $1172$ | $2949$ | $1982$ |  |  |  |
| $5$ | $1$ | $22$ | $226$ | $1172$ | $3069$ | $3782$ |  |  |  |
| $6$ | $1$ | $22$ | $226$ | $1172$ | $3069$ | $4142$ | $2040$ |  |  |
| $7$ | $1$ | $22$ | $226$ | $1172$ | $3069$ | $4142$ | $2760$ |  |  |
| $\geq 8$ | $1$ | $22$ | $226$ | $1172$ | $3069$ | $4142$ | $2760$ | $720$ |  |
$7$ | $2$ | $1$ | $29$ | $413$ | $2863$ | $6446$ |  |  |  |  |
| $3$ | $1$ | $29$ | $420$ | $3248$ | $11507$ | $8651$ |  |  |  |
| $4$ | $1$ | $29$ | $420$ | $3290$ | $13327$ | $22525$ |  |  |  |
| $5$ | $1$ | $29$ | $420$ | $3290$ | $13537$ | $28825$ | $18186$ |  |  |
| $6$ | $1$ | $29$ | $420$ | $3290$ | $13537$ | $29665$ | $32466$ |  |  |
| $7$ | $1$ | $29$ | $420$ | $3290$ | $13537$ | $29665$ | $34986$ | $15960$ |  |
| $8$ | $1$ | $29$ | $420$ | $3290$ | $13537$ | $29665$ | $34986$ | $21000$ |  |
| $\geq 9$ | $1$ | $29$ | $420$ | $3290$ | $13537$ | $29665$ | $34986$ | $21000$ | $5040$ |
$8$ | $2$ | $1$ | $37$ | $714$ | $7434$ | $32709$ | $25953$ |  |  |  |
| $3$ | $1$ | $37$ | $722$ | $8050$ | $46205$ | $95161$ |  |  |  |
| $4$ | $1$ | $37$ | $722$ | $8106$ | $49845$ | $150657$ | $108232$ |  |  |
| $5$ | $1$ | $37$ | $722$ | $8106$ | $50181$ | $167457$ | $253720$ |  |  |
| $6$ | $1$ | $37$ | $722$ | $8106$ | $50181$ | $169137$ | $310840$ | $184464$ |  |
| $7$ | $1$ | $37$ | $722$ | $8106$ | $50181$ | $169137$ | $317560$ | $312144$ |  |
| $8$ | $1$ | $37$ | $722$ | $8106$ | $50181$ | $169137$ | $317560$ | $332304$ | $141120$ |
| $9$ | $1$ | $37$ | $722$ | $8106$ | $50181$ | $169137$ | $317560$ | $332304$ | $181440$ |
| $\geq 10$ | $1$ | $37$ | $722$ | $8106$ | $50181$ | $169137$ | $317560$ | $332304$ | $181440$ | $40320$
Table 3. Nonzero Betti numbers $\beta_{k}$ of
$X^{\mathbb{R}}_{\mathds{H}\mathrm{och}(m,n)}$ up to $m=8$

###  6.5. Cyclohedra

For $n\geq 5$, let us consider a cycle graph $C_{n}$ with the vertices
$1,2,\ldots,n$. Let any vertex labeling of $C_{n}$ be given. There are two
neighbors $x_{1}$ and $x_{2}$ of the vertex $1$. Without loss of generality,
assume that $x_{1}>x_{2}$. Choose a neighbor $x_{3}\neq 1$ of the vertex
$x_{2}$.

When $I=\\{1,x_{1},x_{2},x_{3}\\}$, the induced subgraph $C_{n}|_{I}$ of
$C_{n}$ is given by

$x_{1}$$1$$x_{2}$$x_{3}$

.

Since $1<x_{2}<x_{1}$, by similarly way of Example 4.1, we find that there are
exactly three alternating $\mathcal{B}(C_{n})|_{I}$-permutations. In contrast,
the $a$-number $a(C_{n}|_{I})$ of $C_{n}|_{I}$ is given by

$(\\#\text{edges of }C_{n}|_{I})-1$

from [11], and then $\beta_{1}((K_{\mathcal{B}(C_{n})})_{I})=2$ which is
different from the number of alternating
$\mathcal{B}(C_{n})|_{I}$-permutations. Note that $C_{n}$ is not chordal which
implies that $\mathcal{B}(C_{n})$ is not a chordal building set for any vertex
labeling of $C_{n}$.

##  7\. Open problems on the nested set complex

In this section, we aim to propose several interesting problems based on our
study of the nested set complex.

###  7.1. Venustus building sets

Let us briefly summarize all the steps for the main proof. For a connected
chordal building set $\mathcal{B}$ on $[n+1]=\\{1,\ldots,n+1\\}$, we have that

$\displaystyle\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}})$
$\displaystyle=\sum_{\begin{subarray}{c}I\subset[n+1]\\\ \left|I\right|\text{
is
even}\end{subarray}}\widetilde{\beta}_{k-1}((K_{\mathcal{B}})_{I})\quad(\text{by~{}\eqref{Betti_comp}
as a variant of Theorem~{}\ref{Choi-Park2017_torsion}})$
$\displaystyle=\sum_{\begin{subarray}{c}I\subset[n+1]\\\ \left|I\right|\text{
is
even}\end{subarray}}\widetilde{\beta}_{k-1}(K_{\mathcal{B}|_{I}}^{\operatorname{odd}})\quad(\text{required
the flagness of $K_{\mathcal{B}}$, and by Lemma~{}\ref{remove}})$
$\displaystyle=\sum_{\begin{subarray}{c}I\subset[n+1]\\\ \left|I\right|\text{
is
even}\end{subarray}}\widetilde{\beta}_{\left|I\right|-k-2}(K_{\mathcal{B}|_{I}}^{\operatorname{even}})\quad(\text{by~{}\eqref{Alex_dual}
as the Alexander duality})$
$\displaystyle=\sum_{\begin{subarray}{c}I\subset[n+1]\\\ \left|I\right|\text{
is
even}\end{subarray}}\widetilde{\beta}_{\left|I\right|-k-2}(\Delta(\mathcal{P}_{\mathcal{B}|_{I}}))\quad(\text{by
Lemma~{}\ref{geo_real}})$ $\displaystyle=\sum_{I\in{[n+1]\choose
2k}}a(\mathcal{B}|_{I}))\quad(\text{required the chordality of $\mathcal{B}$,
and by Theorem~{}\ref{thm:shellability_of_K_P_for_chordal}})$
$\displaystyle=\sum_{I\in{[n+1]\choose 2k}}{\\#\text{alternating
$\mathcal{B}|_{I}$-permutations}}\quad(\text{by Theorem~{}\ref{thm:a(G)}}).$

The one of the important steps in the above proof is Theorem 3.5, which
concerns the shellability of $\Delta(\mathcal{P}_{\mathcal{B}|_{I}})$.
Consequently, the reduced homology of each $(K_{\mathcal{B}})_{I}$ is
concentrated in a single degree, with the degree depending only on the size of
$I$. More precisely, if $|I|=2k$, then only
$\widetilde{\beta}_{k-1}((K_{\mathcal{B}})_{I})$ can be positive, and hence,
each $I$ contributes $\beta_{k}(X_{\mathcal{B}}^{\mathbb{R}})$ of degree $k$,
which is half the cardinality of $I$. This phenomenon is advantageous as it
simplifies the calculation of Betti numbers for real toric manifolds
associated with any chordal building set and allows for a highly structured
homology decomposition. Since this phenomenon does not occur for all building
sets, we would like to designate a specific term to those building sets that
exhibit this phenomenon.

######  Definition 7.1.

A connected building set $\mathcal{B}$ on $[n+1]$ is said to be _venustus_ if,
for each $I\in{[n+1]\choose 2k}$, the reduced homology of
$(K_{\mathcal{B}})_{I}$ is concentrated in the degree $k-1$; that is,

$\widetilde{\beta}_{i-1}((K_{\mathcal{B}})_{I})=0$

for all $i\neq k$.

As mentioned in Section 2.2, the class of venustus building sets contains not
only chordal building sets but also graphical ones, as illustrated in Figure
6.

Building setsChordal building setsGraphical building setsVenustus building
sets Figure 6. Venustus building sets

######  Problem 7.2.

Characterize venustus building sets.

It would be natural to extend this concept to the class of all real toric
manifolds. Let $X^{\mathbb{R}}$ be an $n$-dimensional real toric manifold.
Recall that $X^{\mathbb{R}}$ is completely determined by a pair
$(K,\lambda^{\mathbb{R}})$, where $K$ is an $(n-1)$-dimensional simplicial
complex and $\lambda^{\mathbb{R}}$ is a mod $2$ characteristic map over $K$.
Let $\omega_{i}$ be the element of
$\operatorname{row}\lambda_{\mathcal{B}}^{\mathbb{R}}$ corresponding to the
$i$th row of $\lambda_{\mathcal{B}}^{\mathbb{R}}$ as a matrix for
$i=1,\ldots,n$. Each nonzero element
$\omega=\omega_{i_{1}}+\cdots+\omega_{i_{k}}$ of
$\operatorname{row}\lambda_{\mathcal{B}}^{\mathbb{R}}$ can be regarded as a
subset $\\{i_{1},\ldots,i_{k}\\}$ of $2^{[n]}$. If there is a map
$\sigma\colon 2^{[n]}\to[0,n]$ such that

$\widetilde{\beta}_{i-1}(K_{\omega})=0\quad\text{for all
$i\neq\sigma(\omega)$},$

then we say that the (rational) homology group $H^{\ast}(X^{\mathbb{R}})$ of
$X^{\mathbb{R}}$ is _concentrated in $\sigma$_. In particular, a building set
$\mathcal{B}$ on $[n+1]$ is venustus if and only if the homology of
$X^{\mathbb{R}}_{\mathcal{B}}$ is concentrated in $\sigma_{n}$ defined by

$\sigma_{n}(\omega)=\begin{cases}\frac{\left|\omega\right|}{2},&\mbox{if
$\left|\omega\right|$ is even},\\\ \frac{\left|\omega\right|+1}{2},&\mbox{if
$\left|\omega\right|$ is odd},\end{cases}$

for each $\omega\in 2^{[n]}$. Motivated by this, one can define for a pair
$(K,\lambda^{\mathbb{R}})$ to be _venustus_ if the corresponding real toric
manifold has the homology that is concentrated in $\sigma_{n}$.

The characterization of venustus pairs seems quite challenging. Most pairs
$(K,\lambda^{\mathbb{R}})$ are not venustus, and finding a venustus one is
likely to be difficult. For a pair to be venustus, it undoubtedly needs to
have good combinatorial and topological structures for both $K$ and
$\lambda^{\mathbb{R}}$.

One nontrivial example of venustus pairs is the _real Coxeter variety_
$X_{B_{n}}^{\mathbb{R}}$ of type $B_{n}$. According to [10], the corresponding
complexes in the case of $X_{B_{n}}^{\mathbb{R}}$ are flag and shellable,
which makes the variety venustus. In contrast, for other types, none are
venustus as shown in [9] and [14].

Just as a real permutohedral variety $X_{A_{n}}^{\mathbb{R}}$ is generalized
to a real toric manifold derived from a building set, $X_{B_{n}}^{\mathbb{R}}$
can also be extended to ones associated with building sets, as studied in
[19], [32], and [44]. It would also be interesting to ask the characterization
of _type $B$ venustus_ building sets. More generally, we propose the following
problem.

######  Problem 7.3.

Characterize venustus pairs $(K,\lambda^{\mathbb{R}})$; that is, the
characterize real toric manifolds whose homology is concentrated in
$\sigma_{n}$.

###  7.2. Unimodality and multiplicative structures

As shown in Section 6, for certain chordal building sets calculated by the
authors so far, it has been observed that the sequence of the Betti numbers of
$X^{\mathbb{R}}_{\mathcal{B}}$ is unimodal. In particular, we can rigorously
prove that the unimodality holds for real permutohedral varieties. Denote by
$a_{2k}$ the $2k$th Euler zigzag number (A000111 in [37]).

######  Lemma 7.4 (Proposition 3.2 in [38]).

The sequences $\left\\{\frac{a_{2k}}{(2k)!}\right\\}_{r\geq 0}$ is strictly
log-concave.

######  Theorem 7.5.

For a fixed positive integer $n$, the sequence
$\left\\{\binom{n+1}{2k}a_{2k}\right\\}_{r\geq 0}$ is log-concave.

###### Proof.

Set $\alpha_{k}=\frac{a_{2k}}{(2k)!}$ and $\beta_{k}=\binom{n+1}{2k}a_{2k}$.
Then we have that

$\displaystyle\frac{\beta_{k}^{2}}{\beta_{k-1}\beta_{k+1}}$
$\displaystyle=\frac{(n+1-2k+2)(n+1-2k+1)\alpha_{k}^{2}}{(n+1-2k)(n+1-2k-1)\alpha_{k-1}\alpha_{k+1}}$
$\displaystyle>\frac{\alpha_{k}^{2}}{\alpha_{k-1}\alpha_{k+1}}$
$\displaystyle>1\quad\text{(by Lemma~{}\ref{prop:a/nfac}).}$

∎

Even if we consider graphical building sets, we have failed to find any
example whose Betti number sequence is not unimodal. Based on these findings,
we propose the following conjecture.

######  Conjecture 7.6.

For a venustus building set $\mathcal{B}$, the sequence
$\\{\beta_{k}(X^{\mathbb{R}}_{\mathcal{B}})\\}_{k\geq 0}$ representing the
rational Betti numbers of the associated real toric manifold is unimodal.

On the other hand, for a chordal building set $\mathcal{B}$, one remarkable
advantage of our result is that the generator of the cohomology ring of
$X^{\mathbb{R}}_{\mathcal{B}}$ can also be expressed by alternating
$\mathcal{B}$-permutations. This allows for a nice combinatorial description
of multiplicative structures in terms of permutations, similar to a real
permutohedral variety [13]. However, there is a key difference. In the case of
real permutohedral varieties, $K_{\mathcal{B}|_{I}}^{\operatorname{odd}}$ is a
pure simplicial complex, whereas $K_{\mathcal{B}|_{I}}^{\operatorname{odd}}$
is not pure in general. This difference makes it difficult to directly apply
the methods from [13] to the case of a chordal building set $\mathcal{B}$.
Therefore, we leave the following problem.

######  Problem 7.7.

For a connected chordal building set $\mathcal{B}$, give an explicit
description of a multiplicative structure of rational cohomology ring
$H^{\ast}(X^{\mathbb{R}}_{\mathcal{B}})$ in terms of alternating
$\mathcal{B}$-permutations.

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