Frankl–Füredi–Kalai Inequalities on the γ-Vectors of Flag Nestohedra

Abstract

For any flag nestohedron, we define a flag simplicial complex whose f-vector is the γ-vector of the nestohedron. This proves that the γ-vector of any flag nestohedron satisfies the Frankl–Füredi–Kalai inequalities, partially solving a conjecture by Nevo and Petersen (Discrete Comput. Geom. 45:503–521, 2010). We also compare these complexes to those defined by Nevo and Petersen (Discrete Comput. Geom. 45:503–521, 2010) for particular flag nestohedra.

1 Introduction

For any building set $\mathcal{B}$ there is an associated simple polytope P _B called the nestohedron (see Sect. 2, [10, Sect. 7] and [11, Sect. 6]). When $\mathcal{B} = \mathcal {B}(G)$ is the building set determined by a graph G, $P_{\mathcal {B}(G)}$ is the well-known graph-associahedron of G (see [1, Ex. 2.1], [11, Sects. 7 and 12], and [12]). The numbers of faces of $P_{\mathcal{B}}$ of each dimension are conveniently encapsulated in its γ-polynomial $\gamma(\mathcal{B}) =\gamma (P_{\mathcal{B}})$ defined below.

Recall that for a (d−1)-dimensional simplicial complex Δ, the f-polynomial is a polynomial in $\mathbb{Z}[t]$ defined as follows:

$$f(\Delta) (t):= f_0 + f_1t + \cdots+f_{d}t^{d}, $$

where f _i =f _i (Δ) is the number of (i−1)-dimensional faces of Δ, and f ₀(Δ)=1. The h-polynomial is given by

$$h(\Delta) (t): =(t-1)^{d}f(\Delta) \biggl(\frac{1}{t-1} \biggr) = h_0 + h_1t+ \cdots+h_dt^d, $$

where h _i =h _i (Δ). When Δ is a homology sphere, h(Δ) is symmetric, i.e. h _i (Δ)=h _d−i (Δ) for all i (this is known as the Dehn–Sommerville relations); hence it can be written

$$h(\Delta) (t) = \sum_{i=0}^{\lfloor\frac{d}{2}\rfloor} \gamma_it^i(1+t)^{d-2i}, $$

for some $\gamma_{i} \in\mathbb{Z}$. Then the γ-polynomial is given by

$$\gamma(\Delta) (t): = \gamma_0 + \gamma_1t + \cdots+ \gamma_{\lfloor \frac{d}{2}\rfloor}t^{\lfloor\frac{d}{2}\rfloor}, $$

where γ _i =γ _i (Δ). The vectors of coefficients of the f-polynomial, h-polynomial and γ-polynomial are known respectively as the f-vector, h-vector and γ-vector. If P is a simple (d+1)-dimensional polytope then the dual simplicial complex Δ _P of P is the boundary complex (of dimension d) of the polytope that is polar dual to P. The f-vector, h-vector and γ-vector of P are defined via Δ _P as

$$f(P) (t):= t^{d}f(\Delta_P) \bigl(t^{-1}\bigr), $$

so that f _i (P) is the number of i-dimensional faces of P, and

$$\begin{aligned} h(P) (t) :=& h(\Delta_P) (t), \\ \gamma(P) (t) :=& \gamma(\Delta_P) (t). \end{aligned}$$

When $\mathcal{B}$ is a building set, we denote the γ-polynomial for $P_{\mathcal{B}}$ by $\gamma(\mathcal{B})$.

Recall that a simplicial complex Δ is flag if every set of pairwise adjacent vertices is a face. Gal [7] conjectured that:

Conjecture 1.1

[7, Conjecture 2.1.7]

If Δ is a flag homology sphere then γ(Δ) is nonnegative.

This implies that the γ-vector of any flag polytope has nonnegative entries. Gal’s conjecture was proven for flag nestohedra by Volodin in [12, Theorem 9].

In [6] Frankl, Füredi and Kalai characterize the f-vectors of balanced simplicial complexes, and their defining conditions are known as the Frankl–Füredi–Kalai inequalities. Frohmader [5] showed that the f-vector of any flag simplicial complex is the f-vector of a balanced complex. Nevo and Petersen conjectured the following strengthening of Gal’s conjecture:

Conjecture 1.2

[8, Conjecture 6.3]

If Δ is a flag homology sphere then γ(Δ) satisfies the Frankl–Füredi–Kalai inequalities.

They proved this in [8] for the following classes of flag spheres:

• Δ is a Coxeter complex (including the simplicial complex dual to $P_{\mathcal{B}(K_{n})}$),

• Δ is the simplicial complex dual to an associahedron (=$P_{\mathcal{B}(\mathrm{Path}_{n})}$),

• Δ is the simplicial complex dual to a cyclohedron (=$P_{\mathcal{B}(\mathrm{Cyc}_{n})}$),

• Δ has γ ₁(Δ)≤3,

by showing that the γ-vector of such Δ is the f-vector of a flag simplicial complex. In [9], Conjecture 1.2 is proven for the barycentric subdivision of a simplicial sphere, by showing that the γ-vector is the f-vector of a balanced simplicial complex.

In this paper we prove Conjecture 1.2 for all flag nestohedra:

Theorem 1.3

If $P_{\mathcal{B}}$ is a flag nestohedron, there is a flag simplicial complex $\varGamma(\mathcal{B})$ such that $f(\varGamma(\mathcal{B}))=\gamma (P_{\mathcal{B}})$. In particular, $\gamma(P_{\mathcal{B}})$ satisfies the Frankl–Füredi–Kalai inequalities.

Our construction for $\varGamma(\mathcal{B})$ depends on the choice of a “flag ordering” for $\mathcal{B}$ (see Sect. 3). In the special cases considered by Nevo and Petersen [8] our $\varGamma (\mathcal{B})$ does not always coincide with the complex they construct.

After completing this paper, the author proved Conjecture 1.2 in the more general context of edge subdivisions in [2]. This result was also proven independently by Volodin in [13] and [14], who had previously shown in [12] that flag nestohedra are a special case of polytopes obtainable from the cube by 2-truncations (see Theorems 2.5 and 2.6). The author and Volodin are currently working on amalgamating the two results. The result in [2] is shown to be equivalent to the result in this paper for flag nestohedra, where a flag ordering in this context corresponds to a subdivision sequence in [2].

Here is a summary of the contents of this paper. Section 2 contains preliminary definitions and results relating to building sets and nestohedra. In Sect. 3 we define the flag simplicial complex $\varGamma (\mathcal{B})$ for a building set $\mathcal{B}$ and prove Theorem 1.3. In Sect. 4 we compare the simplicial complexes $\varGamma (\mathcal{B})$ to the flag simplicial complexes defined in [8].

2 Preliminaries

A building set $\mathcal{B}$ on a finite set S is a set of nonempty subsets of S such that:

• For any $I, J \in\mathcal{B}$ such that I∩J≠∅, $I \cup J \in\mathcal{B}$.

• $\mathcal{B}$ contains the singletons {i}, for all i∈S.

$\mathcal{B}$ is connected if it contains S. For any building set $\mathcal{B}$, $\mathcal{B}_{\mathrm{max}}$ denotes the set of maximal elements of $\mathcal{B}$ with respect to inclusion. The elements of $\mathcal{B}_{\mathrm{max}}$ form a disjoint union of S, and if $\mathcal{B}$ is connected then $\mathcal{B}_{\mathrm{max}} = \{S\}$. Building sets $\mathcal {B}_{1}$, $\mathcal{B}_{2}$ on S are equivalent, denoted $\mathcal {B}_{1} \cong\mathcal{B}_{2}$, if there is a permutation σ:S→S that induces a one to one correspondence $\mathcal{B}_{1} \rightarrow\mathcal{B}_{2}$.

Example 2.1

Let G be a graph with no loops or multiple edges, with n vertices labelled distinctly from [n]. Then the graphical building set $\mathcal{B}(G)$ is the set of subsets of [n] such that the induced subgraph of G is connected (see [3, 4], [11, Sects. 7 and 12] and [12]). $\mathcal{B}(G)_{\mathrm{max}}$ is the set of connected components of G.

Let $\mathcal{B}$ be a building set on S and I⊆S. The restriction of $\mathcal{B}$ to I is the building set

$$\mathcal{B}|_{I}: = \{J \mid J \subseteq I, \hbox{and }J \in\mathcal {B} \}\quad\hbox{on $I$}. $$

The contraction of $\mathcal{B}$ by I is the building set

$$\mathcal{B}/I: = \bigl\{ J-(J \cap I)\mid J \in\mathcal{B}, J \nsubseteq I \bigr\} \quad\hbox{on $S-I$.} $$

We associate a polytope to a building set as follows. Let e ₁,…,e _n denote the standard basis vectors in $\mathbb{R}^{n}$. Given I⊆[n], define the simplex Δ _I :=ConvexHull(e _i ∣i∈I). Let $\mathcal{B}$ be a building set on [n]. The nestohedron $P_{\mathcal{B}}$ is a polytope defined in [10] and [11] as the Minkowski sum,

$$P_\mathcal{B}:= \sum_{I \in\mathcal{B}} \Delta_I. $$

A (d−1)-dimensional face of a d-dimensional polytope is called a facet. A simple polytope P is flag if any collection of pairwise intersecting facets has nonempty intersection, i.e. its dual simplicial complex is flag. We use the abbreviation flag complex in place of flag simplicial complex. A building set $\mathcal{B}$ is flag if $P_{\mathcal{B}}$ is flag.

A minimal flag building set $\mathcal{D}$ on a set S is a connected building set on S that is flag, such that no proper subset of its elements forms a connected flag building set on S. Minimal flag building sets are described in detail in [11, Sect. 7.2]. They correspond to plane binary trees with leaf set S. Given such a tree, the leaves are labelled 1 to n, and the corresponding minimal flag building set is the union of the set of leaf descendants of each vertex of the tree. If $\mathcal{D}$ is a minimal flag building set then $\gamma(\mathcal{D}) = 1$ (see [11, Sect. 7.2]).

Let $\mathcal{B}$ be a building set. A binary decomposition or decomposition of a non-singleton element $B \in\mathcal{B}$ is a set $\mathcal{D} \subseteq\mathcal{B}$ that forms a minimal flag building set on B. Suppose that $B \in\mathcal{B}$ has a binary decomposition $\mathcal{D}$. The two maximal elements $D_{1},D_{2} \in \mathcal{D}-\{B\}$ with respect to inclusion are the maximal components of B in $\mathcal{D}$. Propositions 2.2 and 2.3 give alternative characterizations of when a building set is flag.

Proposition 2.2

[1, Lemma 7.2]

A building set $\mathcal{B}$ is flag if and only if every non-singleton $B \in\mathcal{B}$ has a binary decomposition.

Proposition 2.3

[1, Corollary 2.6]

A building set $\mathcal{B}$ is flag if and only if for every non-singleton $B \in\mathcal{B}$, there exist two elements $D_{1},D_{2} \in\mathcal{B}$ such that D ₁∩D ₂=∅ and D ₁∪D ₂=B.

It follows from Proposition 2.3 that a graphical building set is flag.

Lemma 2.4

[1, Lemma 2.7]

Suppose $\mathcal{B}$ is a flag building set. If $A,B \in\mathcal{B}$ and A⊆̷B, then there is a decomposition of B in $\mathcal{B}$ that contains A.

Recall the following theorems:

Theorem 2.5

[12, Lemma 6]

Let $\mathcal{B}$ and $\mathcal{B}'$ be connected flag building sets on S such that $\mathcal{B} \subseteq\mathcal{B}'$. Then $\mathcal{B}'$ can be obtained from $\mathcal{B}$ by successively adding elements so that at each step the set is a flag building set.

Theorem 2.6

[7, Proposition 2.4.3], [12, Proposition 3]

If $\mathcal{B}'$ is a flag building set on S obtained from a flag building set $\mathcal{B}$ on S by adding an element I, then

$$\begin{aligned} \gamma\bigl(\mathcal{B}'\bigr) =& \gamma(\mathcal{B}) + t\gamma \bigl(\mathcal{B}'|_I\bigr)\gamma\bigl( \mathcal{B}'/I\bigr) \\ =& \gamma(\mathcal{B}) + t\gamma(\mathcal{B}|_I)\gamma( \mathcal{B}/I). \end{aligned}$$

3 The Flag Complex $\varGamma(\mathcal{B})$ of a Flag Building Set $\mathcal{B}$

In [12], Corollary 5 (which is attributed to Erokhovets [4]) states that any nestohedron $P_{\mathcal{B}}$ is combinatorially equivalent to a nestohedron $P_{\mathcal{B}_{1}}$ for a connected building set $\mathcal{B}_{1}$. Hence to prove Theorem 1.3 we need only consider connected building sets.

Suppose that $\mathcal{B}$ is a connected flag building set on [n], $\mathcal{D}$ is a decomposition of [n] in $\mathcal{B}$, and I ₁,I ₂,…,I _k is an ordering of $\mathcal{B}-\mathcal{D}$, such that $\mathcal{B}_{j}=\mathcal{D}\cup\{I_{1},I_{2},\ldots,I_{j}\}$ is a flag building set for all 0≤j≤k (such an ordering exists by Theorem 2.5). We call the pair consisting of such a decomposition $\mathcal{D}$ and the ordering on $\mathcal{B}-\mathcal {D}$, a flag ordering of $\mathcal{B}$, denoted O, or $(\mathcal{D},I_{1},\ldots,I_{k})$. For any $I_{j} \in\mathcal{B} -\mathcal {D}$, we say an element in $\mathcal{B}_{j-1}$ is earlier in the flag ordering than I _j , and an element in $\mathcal{B} - \mathcal {B}_{j}$ is later in the flag ordering than I _j .

For any j∈[k], define:

$$U_j: =\{i\mid i<j, I_i \nsubseteq I_j, \hbox{there is no } I \in \mathcal{B}_{i-1}\hbox{ such that }I \backslash I_j = I_i \backslash I_j\}, $$

and

$$V_j: = \{i\mid i<j, I_i \subseteq I_j, \hbox{there exists } I \in \mathcal{B}_{i-1} \hbox{ such that }I_i \subsetneq I \subsetneq I_j \}. $$

If i∈U _j ∪V _j then we say that I _i is non-degenerate with respect to I _j . If $I_{i} \in\mathcal{B}_{j-1}$ and i∉U _j , then I _i is U-degenerate with respect to I _j , and if I _i ∉∪V _j then I _i is V-degenerate with respect to I _j .

Given a flag building set $\mathcal{B}$ with flag ordering $O = (\mathcal{D},I_{1},\ldots,I_{k})$ define a graph on the vertex set

$$V_O=\bigl\{ v(I_1),\ldots,v(I_k)\bigr\} , $$

where for any i<j, v(I _i ) is adjacent to v(I _j ) if and only if i∈U _j ∪V _j . Then define a flag simplicial complex Γ(O) whose faces are the cliques in this graph. If the flag ordering is clear then we denote Γ(O) by $\varGamma(\mathcal{B})$. For any S⊆[k], we let Γ(O)| _S denote the induced subcomplex of Γ(O) on the vertices v(I _i ) for all i∈S.

Example 3.1

Consider the flag building set $\mathcal{B}(\mathrm{Path}_{5})$ on [5]. It has a flag ordering O given by

$$\mathcal{D} = \bigl\{ \{1\},\{2\},\{3\},\{4\},\{5\},[2],[3],[4],[5]\bigr\} , $$

and

$$\begin{aligned} I_1 =& \{3,4\},\qquad I_2 = \{2,3,4\},\qquad I_3 = \{2,3\},\\ I_4 =&\{2,3,4,5\},\qquad I_5 = \{ 3,4,5\},\qquad I_6 = \{4,5\}. \end{aligned}$$

Then Γ(O) has only two edges, namely

$$\bigl\{ v(I_2),v(I_6)\bigr\} \quad\hbox{and}\quad\bigl\{ v(I_3),v(I_4)\bigr\} . $$

These are edges because I ₂={2,3,4} is the earliest element which has image {2,3} in the contraction by I ₆, and the element I ₃={2,3} is a subset of I ₂={2,3,4} which is in turn a subset of I ₄.

Suppose that $(\mathcal{D},I_{1},\ldots,I_{k})$ is a flag ordering. Then $\mathcal{D}/I_{k}$ is a decomposition of [n]−I _k , and we have an induced ordering of $(\mathcal{B}/I_{k})-(\mathcal{D}/I_{k})$, where the ith element is $I_{u_{i}}':=I_{u_{i}} \backslash I_{k}$ if u _i is the ith element of U _k (listed in increasing order). Then for all i, $\mathcal{D}/I_{k} \cup\{I_{u_{1}}',\ldots,I_{u_{i}}'\}$ is a flag building set. Hence we can also define a flag complex $\varGamma(\mathcal{B}/I_{k})$. We label the vertices of $\varGamma(\mathcal{B}/I_{k})$ by $v(I_{u_{1}}'),v(I_{u_{2}}'),\ldots,v(I_{u_{|U_{k}|}}')$. Hence, we see that U-degenerate elements with respect to I _j are the elements that do not contribute to the building set $\mathcal{B}_{j}/I_{j}$.

Claim 3.2

Let $\mathcal{B}$ be a connected flag building set with flag ordering $(\mathcal{D},I_{1},\ldots,I_{k})$. For all $I \in\mathcal{B}$ let I′=I∖I _k . Suppose j∈U _k and $I \in\mathcal{B}_{j-1}$. Then I⊆I _j if and only if $I' \subseteq I_{j}'$.

Proof

⇒: It is clear that I⊆I _j implies $I' \subseteq I_{j}'$.

⇐: Suppose for a contradiction that $I' \subseteq I_{j}'$ and I⊈I _j . Then I∩I _j ≠∅ and I∪I _j ≠I _j , which implies that (since $\mathcal{B}_{j}$ is a building set) $I \cup I_{j} \in\mathcal{B}_{j-1}$. We also have that $(I \cup I_{j})' =I_{j}'$, which implies that I _j is U-degenerate with respect to I _k ; a contradiction. □

Proposition 3.3

Let $\mathcal{B}$ be a connected flag building set with flag ordering given by $(\mathcal{D},I_{1},\ldots,I_{k})$. Then $\varGamma(\mathcal{B}/I_{k}) \cong\varGamma(\mathcal{B})|_{U_{k}}$. The map on the vertices is given by $v(I_{i}') \mapsto v(I_{i})$.

Proof

$\varGamma(\mathcal{B})|_{U_{k}}$ is a flag complex with vertex set $v(I_{u_{1}}), v(I_{u_{2}}),\ldots,v(I_{u_{|U_{k}|}})$ and $\varGamma(\mathcal {B}/I_{k})$ is a flag complex with vertex set $v(I_{u_{1}}'), v(I_{u_{2}}'),\ldots,v(I_{u_{|U_{k}|}}')$. Suppose that i<j where i,j∈U _k . We need to show that $\{v(I_{j}'),v(I_{i}')\} \in\varGamma(\mathcal {B}/I_{k})$ if and only if $\{v(I_{j}),v(I_{i})\} \in\varGamma(\mathcal {B})|_{U_{k}}$. Note that by Claim 3.2, I _i ⊆I _j if and only if $I_{i}' \subseteq I_{j}'$.

(1) Suppose that I _i ⊆I _j , and that $\{v(I_{i}'),v(I_{j}')\} \in\varGamma(\mathcal{B}/I_{k})$, so that there exists $I \in\mathcal {B}_{i-1}$ such that $I_{i}' \subsetneq I' \subsetneq I_{j}'$. By Claim 3.2, I⊆I _j and since I _i ⊆I _j this implies I∪I _i ⊆I _j . Since I∩I _i ≠∅, we have $I \cup I_{i} \in\mathcal{B}_{i-1}$. Hence I _i ⊆̷I∪I _i ⊆̷I _j which implies $\{ v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal{B})|_{U_{k}}$.

Suppose that I _i ⊆I _j and that $\{v(I_{i}),v(I_{j})\} \in \varGamma(\mathcal{B})|_{U_{k}}$, so that there exists $I \in\mathcal {B}_{i-1}$ such that I _i ⊆̷I⊆̷I _j . Then $I_{i}' \subseteq I' \subseteq I_{j}'$, and $I' \ne I_{i}'$ and $I' \ne I_{j}'$ since i,j∈U _k , so that $I_{i}' \subsetneq I' \subsetneq I_{j}'$. Hence $\{v(I_{i}'),v(I_{j}')\} \in\varGamma(\mathcal{B}/I_{k})$.

(2) Suppose that I _i ⊈I _j , and that $\{ v(I_{i}'),v(I_{j}')\} \in\varGamma(\mathcal{B}/I_{k})$, and suppose for a contradiction that $\{v(I_{i}),v(I_{j})\} \notin\varGamma(\mathcal {B})|_{U_{k}}$, i.e. i∉U _j . Then there exists $I \in\mathcal {B}_{i-1}$ such that I∖I _j =I _i ∖I _j . Then $I' \backslash I_{j}' = I_{i}' \backslash I_{j}'$ which implies the contradiction that $\{v(I_{i}'),v(I_{j}')\} \notin\varGamma(\mathcal {B}/I_{k})$.

Suppose that I _i ⊈I _j , and that $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal{B})|_{I_{k}}$. We will prove the contrapositive that $\{v(I_{i}'),v(I_{j}')\} \notin\varGamma(\mathcal{B}/I_{k})$ implies that $\{v(I_{i}),v(I_{j})\} \notin\varGamma(\mathcal{B})|_{U_{k}}$. $\{ v(I_{i}'),v(I_{j}')\} \notin\varGamma(\mathcal{B}/I_{k})$ implies there exists $M \in\mathcal{B}_{i-1}$ such that $M'\backslash I_{j}' = I_{i}'\backslash I_{j}'$.

• Assume that M⊆I _i , and for this case refer to Fig. 1. Let R:=(I _i ∖(M∪I _j )), and note that this is a subset of I _k since I _i ∖(M∪I _i ∪I _k )=∅ by the definition of M. Also, let J:=I _i ∖(M∪I _k ). Since M⊆I _i , by Lemma 2.4, there exists a decomposition of I _i in $\mathcal {B}_{i}$ that contains M. Hence M is contained in a maximal component D of this decomposition. Let D′ be the other maximal component, and note that D∩D′=∅. If D′∩R=∅ then $\{v(I_{i}),v(I_{j})\} \notin\varGamma(\mathcal {B})|_{U_{k}}$ since D∖I _j =I _i ∖I _j , hence the desired condition holds. If D′∩J=∅ then I _i ∖I _k =D∖I _k which contradicts i∈U _k . If D′∩J≠∅ and D′∩R≠∅, then (D′∪I _j )∖I _k =I _j ∖I _k , which contradicts j∈U _k .

  Fig. 1

  

  A picture of the sets in case (2), assuming M⊆I _i . Note that I _i ∖(M∪I _j ∪I _k )=∅ by the definition of M

• Assume that M⊈I _i . For this case refer to Fig. 2. Let H:=I _i ∖(I _j ∪I _k ). In $(\mathcal{B}_{j}/I_{k})/I_{j}'$ both $I_{i}'$ and M′ have the same image that is given by H, and H≠∅ since H=∅ implies $I_{i}' \subseteq I_{j}'$, which contradicts Claim 3.2. Let K:=M∖(I _k ∪I _i ). Then K≠∅ since K=∅ implies I _i ∖I _k =M∖I _k , which contradicts i∈U _k . Let L:=M∖(I _i ∪I _j ). L=∅ implies $\{v(I_{i}),v(I_{j})\} \notin\varGamma(\mathcal{B})|_{U_{k}}$ since (I _i ∪M)∖I _j =I _i ∖I _j , so the desired condition holds. Suppose now L≠∅. Then M intersects each of H,K and L. Let I be a minimal (for inclusion) element in $\mathcal{B}_{i-1}$ that intersects H,K and L. Then |I|≥3 and at least one of the maximal components of a decomposition of I (in $\mathcal{B}_{i-1}$) must intersect exactly two of K,H and L (since I is minimal with respect to intersecting H, K and L, and the components cannot both intersect exactly one set since their disjoint union is I). Denote such an element by $\widehat{D}$. Note that since $\widehat{D} \in\mathcal{B}_{i-1}$, and $\widehat{D} \cap I_{i} \ne\emptyset$, this implies by the definition of a building set that $\widehat{D} \cup I_{i} \in\mathcal{B}_{i-1}$. If $\widehat{D}$ intersects K and L then $(I_{j} \cup\widehat{D}) \backslash I_{k} = I_{j} \backslash I_{k}$ which contradicts j∈U _k . If $\widehat{D}$ intersects both K and H then $\{v(I_{i}),v(I_{j})\} \notin \varGamma(\mathcal{B})|_{U_{k}}$ since $(I_{i} \cup\widehat{D}) \backslash I_{j} = I_{i} \backslash I_{j}$, so the desired condition holds. If $\widehat{D}$ intersects L and H, then $(I_{i} \cup\widehat{D}) \backslash I_{k} =I_{i} \backslash I_{k}$, which contradicts i∈U _k .

  Fig. 2

  

  A picture of the sets in case (2), assuming M⊈I _i . Note that I _i ∖(M∪I _j ∪I _k )=∅ by the definition of M

 □

We now consider the flag building set $\mathcal{B}|_{I_{k}}$. It is not necessarily true that $\mathcal{D}|_{I_{k}}$ is a decomposition of I _k . Let

$$\mathcal{D}_k: = \mathcal{D}|_{I_k} \cup\{I_j \mid I_j \subseteq I_k, j \notin V_k\}. $$

The following claim holds for $\mathcal{D}_{k}$.

Claim 3.4

Suppose $\mathcal{B}$ is a flag building set with flag ordering $(\mathcal{D},I_{1},\ldots,I_{k})$. Then $\mathcal{D}_{k}$ is a decomposition of I _k in $\mathcal{B}|_{I_{k}}$, and for any i≤k, $\mathcal{D}_{k} \cup\{I_{i}\mid i \le j\emph{ and }i \in V_{k} \}$ is a flag building set on I _k .

Proof

We will first show that $\mathcal{D}_{k}$ is a decomposition of I _k in $\mathcal{B}|_{I_{k}}$. This can be seen by induction. We assume that for some i<k, the set of V-degenerate elements with respect to I _k in $\mathcal{B}_{i}$, that are a subset of I _k , together with $\mathcal {D}|_{I_{k}}$, are the union of a decomposition for each element in $(\mathcal{B}_{i}|_{I_{k}})_{\mathrm{max}}$. Then if I _i+1⊆I _k and i+1∉V _k , then I _i+1 is the union of two elements in $(\mathcal{B}_{i}|_{I_{k}})_{\mathrm{max}}$, so that the inductive hypothesis holds for i+1. It is also true that if I _i+1⊆I _k and i+1∈V _k , or if I _i+1⊈I _k , that the inductive hypothesis holds for i+1. The hypothesis clearly holds for i=0. Hence this statement holds by induction.

We will now show that for any i≤k, $\mathcal{D}_{k} \cup\{I_{i}\mid i \le j \hbox{ and } i \in V_{k} \}$ is a flag building set on I _k . This is true since $\mathcal{B}_{i}|_{I_{k}}$ is a flag building set, and each element in $\mathcal{B}_{i}|_{I_{k}}$ is a subset of, or disjoint to any element in $\mathcal{D}_{k} - \mathcal{B}_{i}|_{I_{k}}$. □

Since Claim 3.4 holds, we define $\varGamma(\mathcal{B}|_{I_{k}})$ to be the flag complex Γ(O) with respect to the flag ordering O of $\mathcal{B}|_{I_{k}}$ with decomposition $\mathcal{D}_{k}$ and ordering of $\mathcal{B}|_{I_{k}} - \mathcal{D}_{k}$ given by $I_{v_{1}},I_{v_{2}},\ldots,I_{v_{|V_{k}|}}$ where v _j is the jth element of V _k listed in increasing order. We label the vertices of $\varGamma (\mathcal{B}|_{I_{k}})$ by $v(I_{v_{1}}),\ldots,v(I_{u_{|V_{k}|}})$ rather than by their index in V _k . In keeping with the notation that $\mathcal {B}_{j}$ is the flag building set obtained after adding elements indexed up to j, we let $(\mathcal{B}|_{I_{k}})_{j}$ denote the flag building set $\mathcal{D}_{k} \cup\{I_{i}\mid i \le j \hbox{ and } i \in V_{k}\}$, so that $\varGamma((\mathcal{B}|_{I_{k}})_{j})$ is defined. Note then that for any j, $\mathcal{B}_{j}|_{I_{k}} \subseteq(\mathcal{B}|_{I_{k}})_{j}$.

Proposition 3.5

Let $\mathcal{B}$ be a connected flag building set with flag ordering given by $(\mathcal{D},I_{1},\ldots,I_{k})$. Then $\varGamma(\mathcal{B}|_{I_{k}}) = \varGamma(\mathcal{B})|_{V_{k}}$.

Proof

Both $\varGamma(\mathcal{B}|_{I_{k}})$ and $\varGamma(\mathcal{B})|_{V_{k}}$ are both flag complexes with the vertex set $v(I_{v_{1}}),v(I_{v_{2}}),\ldots,v(I_{u_{|V_{k}|}})$. We need to show that for any i,j∈V _k where i<j, $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal {B})|_{V_{k}}$ if and only if $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal {B}|_{I_{k}})$.

⇒: Suppose that $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal {B})|_{V_{k}}$. First assume that I _i ⊆I _j . Then there is some $I \in\mathcal{B}_{i-1}$ such that I _i ⊆̷b⊆̷I _j . Since $I \in\mathcal{B}_{i-1}|_{I_{k}}$ and $\mathcal {B}_{i-1}|_{I_{k}} \subseteq(\mathcal{B}|_{I_{k}})_{i-1}$ this implies that $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal{B}|_{I_{k}})$.

Now suppose that I _i ⊈I _j . Suppose for a contradiction that $\{v(I_{i}),v(I_{j})\} \notin\varGamma(\mathcal{B}|_{I_{k}})$. Then there exists some $D \in\mathcal{D}_{k}-\mathcal{D}|_{I_{k}}$, $D \in \mathcal{B}_{i-1}$, such that D∪I _j =I _i ∪I _j . Since i∈V _k , there exists some $I \in\mathcal{B}_{i-1}$ such that I _i ⊆̷I⊆̷I _k . Since $\{v(I_{i}),v(I_{j})\} \in\varGamma (\mathcal{B})|_{V_{k}}$, we have that I∖(I _i ∪I _j )≠∅. Since the index of D is not in V _k , every element in the restriction to I _k that is earlier than D in the flag ordering is a subset of it or does not intersect it. This implies I⊆D, so D∖(I _i ∪I _j )≠∅, which contradicts D∪I _j =I _i ∪I _j .

⇐: Suppose that $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal {B}|_{I_{k}})$. First assume that I _i ⊆I _j , so that there is some $D \in(\mathcal{B}|_{I_{k}})_{i-1}$ such that I _i ⊆̷D⊆̷I _j . If $D \in\mathcal{B}_{i-1}|_{I_{k}}$ then clearly $\{ v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal{B})|_{V_{k}}$, as desired. If $D \notin\mathcal{B}_{i-1}|_{I_{k}}$ then $D \in\mathcal{D}_{k} - \mathcal {D}|_{I_{k}}$. Since i∈V _k , there exists some $I \in\mathcal {B}_{i-1}$ such that I _i ⊆̷I⊆̷I _k . Since the index of D is not in V _k , we have that I _i ⊆̷I⊆̷D. This is because D either contains or does not intersect elements that are earlier in the flag ordering and contained in I _k . Then since D⊆̷I _j this implies I⊆̷I _j and since $I \in \mathcal{B}_{i-1}$ and I _i ⊆̷I⊆̷I _j , this implies $\{v(I_{i}),v(I_{j})\} \in\varGamma(\mathcal{B})|_{V_{k}}$.

Now assume that I _i ⊈I _j . Suppose for a contradiction that $\{v(I_{i}),v(I_{j})\} \notin\varGamma(\mathcal{B})|_{V_{k}}$. Then there exists $I \in\mathcal{B}_{i-1}|_{I_{k}}$ such that I∪I _j =I _i ∪I _j . Since $\mathcal{B}_{i-1}|_{I_{k}} \subseteq(\mathcal {B}|_{I_{k}})_{i-1}$, this contradicts $\{v(I_{i}),v(I_{j})\} \in\varGamma (\mathcal{B}|_{I_{k}})$. □

Theorem 3.6

Let $\mathcal{B}$ be a connected flag building set with flag ordering O. Then $\gamma(\mathcal{B}) = f(\varGamma(O))$.

Proof

This is a proof by induction on the number of elements of $\mathcal {B}-\mathcal{D}$, and on the size of the set S that $\mathcal{B}$ is on. The result holds for k=0 since $f(\varGamma(\mathcal{D})) = 1 = \gamma(\mathcal{D})$, and when |S|=1. So we assume k≥1 and that the result holds for all connected flag building sets with a smaller value of k.

By Propositions 3.3 and 3.5 and the inductive hypothesis we have $f(\varGamma(\mathcal{B})|_{U_{k}}) = f(\varGamma(\mathcal {B}/{I_{k}}))= \gamma(\mathcal{B}/I_{k})$, and $f(\varGamma(\mathcal {B})|_{V_{k}}) = f(\varGamma(\mathcal{B}|_{I_{k}}))= \gamma(\mathcal {B}|_{I_{k}})$.

Suppose that u∈U _k and w∈V _k . Then $\{v(I_{u}),v(I_{w})\} \in \varGamma(\mathcal{B})$, for suppose, by way of contradiction, that $\{ v(I_{u}),v(I_{w})\} \notin\varGamma(\mathcal{B})$, and suppose that u<w. Then there is some element $I \in\mathcal{B}_{u-1}$ such that I∪I _w =I _u ∪I _w . This implies that I∪I _k =I _u ∪I _k , which contradicts u∈U _k . Suppose that w<u. Then either I _u ∩I _w =∅ or I _w ⊆I _u (otherwise I _u ∪I _w makes I _u U-degenerate with respect to I _k ). Suppose that I _w ∩I _u =∅. Then since $\{v(I_{u}),v(I_{w})\} \notin\varGamma (\mathcal{B})$, there exists $I \in\mathcal{B}_{w-1}$ such that I∪I _u =I _w ∪I _u , and I∩I _u ≠∅. Then I∪I _u makes I _u U-degenerate with respect to I _k ; a contradiction. Suppose that I _w ⊆I _u . Now w∈V _k implies there is some $I \in\mathcal{B}_{w-1}$ such that I _w ⊆̷I⊆̷I _k . Also, I⊆I _u else I∪I _u makes I _u U-degenerate with respect to I _k . However, this implies the contradiction that $\{ v(I_{u}),v(I_{w})\} \in\varGamma(\mathcal{B})$ since I _w ⊆̷I⊆̷I _u .

Hence

$$\varGamma(\mathcal{B})|_{U_k \cup V_k} = \varGamma(\mathcal{B})|_{U_k}* \varGamma (\mathcal{B})|_{V_k}, $$

and therefore

$$f\bigl(\varGamma(\mathcal{B})|_{U_k \cup V_k}\bigr) = f\bigl(\varGamma(\mathcal {B})|_{U_k}\bigr)f\bigl(\varGamma(\mathcal{B})|_{V_k}\bigr)= \gamma(\mathcal{B}/I_k)\gamma (\mathcal{B}|_{I_k}). $$

Since the vertex v(I _k ) is adjacent to the vertices indexed by elements in U _k ∪V _k , we have

$$f\bigl(\varGamma(\mathcal{B})\bigr) = f\bigl(\varGamma(\mathcal{B}_{k-1}) \bigr) + t\gamma(\mathcal {B}/I_k)\gamma(\mathcal{B}|_{I_k}). $$

By the induction hypothesis this implies that

$$f\bigl(\varGamma(\mathcal{B})\bigr) =\gamma(\mathcal{B}_{k-1}) + t \gamma(\mathcal {B}|{I_k})\gamma(\mathcal{B}/I_k), $$

which implies that $f(\varGamma(\mathcal{B})) = \gamma(\mathcal{B})$ by Theorem 2.6. □

For two flag orderings O ₁, O ₂ of a connected flag building set $\mathcal{B}$, it is not necessarily true that the flag complexes Γ(O ₁), Γ(O ₂) are equivalent (up to change of labels on the vertices) even if they have the same decomposition. The following example provides a counterexample.

Example 3.7

Let $\mathcal{B} = \mathcal{B}(\mathrm{Cyc}_{5})$, and let

$$\mathcal{D} =\bigl\{ \{1\},\{2\},\{3\},\{4\},\{5\},[2],[3],[4],[5]\bigr\} . $$

Let O ₁ be the flag ordering with decomposition $\mathcal{D}$ and the following ordering of $\mathcal{B}-\mathcal{D}$:

$$\begin{aligned} &\{2,3\},\ \{2,3,4\},\ \{2,3,4,5\},\ \{4,5\},\ \{3,4,5\},\ \{3,4\}, \\ &\{3,4,5,1\},\ \{4,5,1,2\},\ \{5,1,2,3\},\ \{4,5,1\},\ \{5,1,2\},\ \{1,5\}. \end{aligned}$$

Let O ₂ be the flag ordering with decomposition $\mathcal{D}$ and the following ordering of $\mathcal{B}-\mathcal{D}$:

$$\begin{aligned} &\{2,3\},\ \{2,3,4\},\ \{2,3,4,5\},\ \{3,4\},\ \{3,4,5\},\ \{4,5\},\ \{3,4,5\},\ \{ 3,4\}, \\ &\{3,4,5,1\},\ \{4,5,1,2\},\ \{5,1,2,3\},\ \{4,5,1\},\ \{5,1,2\},\ \{1,5\}. \end{aligned}$$

Then Γ(O ₁) and Γ(O ₂) are depicted in Fig. 3.

Fig. 3

Γ(O ₁) is on the left, and Γ(O ₂) is on the right

4 The Flag Complexes of Nevo and Petersen

In this section we compare the flag complexes that we have defined to those defined for certain graph-associahedra by Nevo and Petersen [8]. They define flag complexes $\varGamma(\widehat{\mathfrak{S}}_{n})$, $\varGamma(\widehat{\mathfrak{S}}_{n}(312))$ and Γ(P _n ) such that:

• $\gamma(\mathcal{B}(K_{n})) = f(\varGamma(\widehat{\mathfrak{S}}_{n}))$,

• $\gamma(\mathcal{B}(\mathrm{Path}_{n})) = f(\varGamma(\widehat{\mathfrak{S}}_{n}(312)))$,

• $\gamma(\mathcal{B}(\mathrm{Cyc}_{n})) = f(\varGamma(P_{n}))$.

In Proposition 4.3, we show that for all n, there is a flag ordering for $\mathcal{B}(\mathrm{Path}_{n})$ so that

$$\varGamma\bigl(\mathcal{B}(\mathrm{Path}_n)\bigr) \cong\varGamma\bigl( \widehat{\mathfrak{S}}_n(312)\bigr). $$

We also show, namely in Propositions 4.2 and 4.5, that the analogous statement is not true for $\mathcal{B}(K_{n})$ and $\mathcal{B}(\mathrm{Cyc}_{n})$, although we have omitted the proofs, which were done by a manual case analysis.

4.1 The Flag Complexes $\varGamma(\mathcal{B}(K_{n}))$ and $\varGamma (\widehat{\mathfrak{S}}_{n})$

The permutohedron is the nestohedron $P_{\mathcal{B}(K_{n})}$. Note that $\mathcal{B}(K_{n})$ consists of all nonempty subsets of [n]. The γ-polynomial of $P_{\mathcal{B}(K_{n})}$ is the descent generating function of $\widehat{\mathfrak{S}}_{n}$, which denotes the set of permutations with no double descents or final descent (see [11, Theorem 11.1]). First we recall the definition of $\varGamma(\widehat {\mathfrak{S}}_{n})$ given by Nevo and Petersen [8, Sect. 4.1].

A peak of a permutation w=w ₁⋯w _n in $\mathfrak{S}_{n}$ is a position i∈[1,n−1] such that w _i−1<w _i >w _i+1, (where w ₀:=0). We denote a peak at position i with a bar w ₁⋯w _i |w _i+1⋯w _n . A descent of a permutation w=w ₁⋯w _n is a position i∈[n−1] such that w _i+1<w _i . Let $\widehat{\mathfrak{S}}_{n}$ denote the set of permutations in $\mathfrak {S}_{n}$ with no double (i.e. consecutive) descents or final descent, and let $\widetilde{\mathfrak{S}}_{n}$ denote the set of permutations in $\mathfrak{S}_{n}$ with one peak. Then $\widehat{\mathfrak{S}}_{n} \cap \widetilde{\mathfrak{S}}_{n}$ consists of all permutations of the form

$$w_1\cdots w_i|w_{i+1}\cdots w_n, $$

where 1≤i≤n−2, w ₁<⋯<w _i , w _i >w _i+1, w _i+1<⋯<w _n .

Define the flag complex $\varGamma(\widehat{\mathfrak{S}}_{n})$ on the vertex set $\widehat{\mathfrak{S}}_{n} \cap\widetilde{\mathfrak{S}}_{n}$ where two vertices

$$u =u_1|u_2 $$

and

$$v = v_1|v_2 $$

with |u ₁|<|v ₁| are adjacent if there is a permutation $w \in \mathfrak{S}_{n}$ of the form

$$w = u_1|a|v_2. $$

Equivalently, if v ₂⊆u ₂, |u ₂−v ₂|≥2, min(u ₂−v ₂)<max(u ₁) and max(u ₂−v ₂)>min(v ₂). (Since there must be two peaks in w this implies |a|≥2.) The faces of $\varGamma (\widehat{\mathfrak{S}}_{n})$ are the cliques in this graph.

Example 4.1

Taking only the part after the peak, $\widehat{\mathfrak{S}}_{5} \cap \widetilde{\mathfrak{S}}_{5}$ can be identified with the set of subsets of [5] of sizes 2,3 and 4 which are not {4,5},{3,4,5}, or {2,3,4,5}. Then the edges of $\varGamma(\widehat{\mathfrak{S}}_{5})$ are given by:

{1,2,3,4} is adjacent to each of {1,2},{1,3},{1,4},{2,3},{2,4},

{1,2,3,5} is adjacent to each of {1,2},{1,3},{1,5},{2,3},{2,5},

{1,2,4,5} is adjacent to each of {1,4},{1,5},{2,4},{2,5}, and

{1,3,4,5} is adjacent to each of {3,4},{3,5}.

Proposition 4.2

There is no flag ordering of $\mathcal{B}(K_{5})$ so that

$$\varGamma\bigl(\mathcal{B}(K_5)\bigr) \cong\varGamma(\widehat{ \mathfrak{S}}_5). $$

The proof of Proposition 4.2, which is a manual case analysis, has been omitted.

4.2 The Flag Complexes $\varGamma(\mathcal{B}(\mathrm{Path}_{n}))$ and $\varGamma (\widehat{\mathfrak{S}}_{n}(312))$

The associahedron is the nestohedron $P_{\mathcal{B}(\mathrm{Path}_{n})}$. Note that $\mathcal{B}(\mathrm{Path}_{n})$ consists of all intervals [j,k] with 1≤j≤k≤n. The γ-polynomial of the associahedron is the descent generating function of $\widehat{\mathfrak{S}}_{n}(312)$, which denotes the set of 312-avoiding permutations with no double or final descents (see [11, Sect. 10.2]). We now describe the flag complex $\varGamma(\widehat{\mathfrak{S}}_{n}(312))$ defined by Nevo and Petersen [8, Sect. 4.2].

Given distinct integers a,b,c,d such that a<b and c<d, the pairs (a,b),(c,d) are non-crossing if either:

• a<c<d<b (or c<a<b<d), or

• a<b<c<d (or c<d<a<b).

Define $\varGamma(\widehat{\mathfrak{S}}_{n}(312))$ to be the flag complex on the vertex set

$$V_{n}: = \bigl\{ (a,b)\mid 1 \le a<b \le n-1\bigr\} , $$

with faces the sets S of V _n such that if (a,b)∈S and (c,d)∈S then (a,b) and (c,d) are non-crossing.

Let O denote the flag ordering of $\mathcal{B} =\mathcal{B}(\mathrm{Path}_{n})$ with decomposition $\mathcal{D}=\{\{1\},\{2\},\{3\},\{4\},\{n\},[2],[3],[4],[n]\}$, where elements $A,B \in\mathcal{B}-\mathcal{D}$ are ordered so that A is earlier than B if:

• max(A)<max(B), or

• max(A)=max(B) and |A|>|B|.

Proposition 4.3

For the flag ordering O of $\mathcal{B} = \mathcal{B}(\mathrm{Path}_{n})$ described above, $\varGamma(O) \cong\varGamma(\widehat{\mathfrak {S}}_{n}(312))$ where the bijection on the vertices is given by v([a+1,b+1])↦(a,b).

Proof

Since $\mathcal{B}-\mathcal{D} =\{[j,k]\mid 2 \le j < k \le n\}$, it is clear that the stated map on vertices is a bijection. Let [l,m],[j,k] be distinct elements of $\mathcal{B}-\mathcal{D}$ with [l,m] occurring before [j,k]. Then m≤k, and if m=k we have l<j. If [l,m]⊈[j,k] then v([l,m]) is adjacent to v([j,k]) if and only if m<j. If [l,m]⊆[j,k] (which entails m<k), then v([l,m]) is adjacent to v([j,k]) if and only if j<l. So in either case v([l,m]) is adjacent to v([j,k]) if and only if (l−1,m−1) and (j−1,k−1) are non-crossing. □

4.3 The Flag Complexes $\varGamma(\mathcal{B}(\mathrm{Cyc}_{n}))$ and Γ(P _n )

The cyclohedron is the nestohedron $P_{\mathcal{B}(\mathrm{Cyc}_{n})}$. Note that $\mathcal{B}(\mathrm{Cyc}_{n})$ consists of all sets {i,i+1,i+2,…,i+s} where i∈[n], s∈{0,1,…,n−1}, and the elements are taken mod n. By [11, Proposition 11.15], $\gamma_{r}(\mathcal{B}(\mathrm{Cyc}_{n})) = \binom {n}{r,r,n-2r}$. We now describe the flag complex Γ(P _n ) defined by Nevo and Petersen [8, Sect. 4.3].

Define the vertex set

$$V_{P_n}:=\bigl\{ (l,r) \in[n-1] \times[n-1]\mid l \ne r\bigr\} . $$

Γ(P _n ) is the flag complex on the vertex set $V_{P_{n}}$ where vertices (l ₁,r ₁),(l ₂,r ₂) are adjacent in Γ(P _n ) if and only if l ₁,l ₂,r ₁,r ₂ are all distinct and either l ₁<l ₂ and r ₁<r ₂, or l ₂<l ₁ and r ₂<r ₁.

Example 4.4

Γ(P ₅) is the flag complex on vertices

$$\begin{aligned} V_{P_5} =&\bigl\{ (1,2),(1,3),(1,4),(2,3),(2,4),(3,4),\\ &\hphantom{\bigl\{ }(2,1),(3,1),(4,1),(3,2),(4,2),(4,3) \bigr\} \end{aligned}$$

with edges

$$\begin{aligned} &\bigl\{ (1,3),(2,4)\bigr\} ,\ \bigl\{ (3,1),(4,2)\bigr\} ,\ \bigl\{ (1,2),(3,4)\bigr\} , \\ &\bigl\{ (1,2),(4,3)\bigr\} ,\ \bigl\{ (2,1),(4,3)\bigr\} ,\ \bigl\{ (2,1),(3,4)\bigr\} . \end{aligned}$$

Note that Γ(P ₅) has exactly two vertices of degree two, and has six connected components, four of which contain more than one vertex.

Proposition 4.5

There is no flag ordering of $\mathcal{B}(\mathrm{Cyc}_{5})$ so that $\varGamma (\mathcal{B}(\mathrm{Cyc}_{5})) \cong\varGamma(P_{5})$.

The proof of Proposition 4.5, which is a manual case analysis, has been omitted.