CELLOHEDRA

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Stephen Robert Reisdorf

May, 2012 CELLOHEDRA

Stephen Robert Reisdorf

Thesis

Approved: Accepted:

Advisor Dean of the College Dr. Stefan Forcey Dr. Chand Midha

Faculty Reader Dean of the Graduate School Dr. James P. Cossey Dr. George R. Newkome

Faculty Reader Date Dr. Jeffrey Riedl

Department Chair Dr. Timothy Norfolk

ii ABSTRACT

The has been generalized to a great variety of combinatorial structures.

In each example the convex is found whose poset is the same as a certain poset structure on the combinatorial structures. Here we find whose face poset models the containment order of certain order ideals of the face poset of a cell complex. This is progress in a program which asks which posets in general have their ideals modeled by convex polytopes.

iii ACKNOWLEDGEMENTS

Thanks to my thesis advisor Dr. Stefan Forcey for countless meetings and invaluable assistance through every phase of the writing process. Thanks to Dr. Jeffery Riedl and Dr. James Cossey for serving as faculty readers, as well as Dr. Stuart Clary for his input it revisions. Thanks to Dr. Gerald Young and Dr. Curtis Clemens for there help with LaTeX.

iv TABLE OF CONTENTS

Page

LISTOFFIGURES ...... vii

CHAPTER

I. INTRODUCTION...... 1

1.1 Thecellcomplexassociahedra ...... 1

1.2 Relatedwork...... 1

1.3 Furtherwork ...... 2

II. MOTIVATINGCONCEPTS ...... 3

2.1 The Classical Combinatorially-Described Polytopes ...... 3

2.2 Generalizing the Classical Polytopes ...... 6

III.DEFINITIONS ...... 11

3.1 Basic Topology of Rn ...... 11

3.2 CellComplexes...... 12

3.3 TubesandTubings...... 12

IV.EXAMPLES...... 14

4.1 SomeThree-DimensionalExamples...... 14

4.2 AFour-DimensionalExample ...... 16

v 4.3 SomeEasyCases...... 16

V. THEOREMS...... 23

5.1 Single Bundles Have a Trivial Effect on KX ...... 23

5.2 KX Can Be Expressed In Terms of Connected Components and theEdgelessGraph ...... 23 5.3 TheTruncationMethod ...... 24

BIBLIOGRAPHY ...... 29

vi LIST OF FIGURES

Figure Page

2.1 Points calculated for the associahedra in 1-dimension and 2-dimensions. 5

2.2 The associahedra in 1-dimension and 2-dimensions...... 6

2.3 The permutohedra in 1-dimension and 2-dimensions...... 7

2.4 The associahedron and permutohedron in 2-dimensions...... 8

2.5 The square as a pseudograph associahedron ...... 9

2.6 An example of a pseudograph and the labeling of ∆G...... 10

2.7 An example of a pseudograph associahedron obtained by truncation. . . 10

3.1 Anexampleofatubeandanon-tube...... 13

4.1 The 3-dimensional cube as a cellohedron ...... 14

4.2 Thehexagonalprismasacellohedron ...... 15

4.3 A4-dimensionalcellohedron ...... 19

4.4 FacetsfromFigure4.3...... 20

4.5 MorefacetsfromFigure4.3...... 21

4.6 The cellohedron of a four cycle filled in with two sheets ...... 22

5.1 An example of labeling a △△X ...... 25

vii 5.2 Vertices are labeled. These labels extend to faces of all dimensions. .. 26

5.3 These three labeled edges correspond to full tubes...... 26

5.4 The preceding three labeled edges have been truncated...... 27

5.5 Now we have truncated the four 2-faces corresponding to non-full tubeswithonlytwonodesandoneedge...... 27 5.6 Finally we have truncated the two 2-faces corresponding to non-full tubeswiththreenodes...... 28

viii CHAPTER I

INTRODUCTION

1.1 The cell complex associahedra

Beginning with the associahedra of James Stasheff in [8], interest in combinatorially described polytopes has developed into an entire field of study. Among such poly- topes are the associahedra, permutohedra, and cyclohedra. These polytopes were generalized to graph associahedra by Michael Carr and Satyan Devadoss in [1], and even further to pseudograph associahedra by the same authors with the addition of

Stefan Forcey in [2]. This paper will construct an even larger class, called cellohedra, for cell complexes which are regular and finite.

1.2 Related work

The graph associahedra, pseudograph associahedra, and cellohedra are only one vein of generalizations of these combinatorially described polytopes. Postnikov, Reiner, and Williams have defined generalized permutohedra which generalize the permuto- hedra defined as the convex hull of the of points in Rn [6]. Zelevinksky described the polytopes related to nested complexes [10]. Algebraic structure has

1 been given to the faces of some of these polytopes. Of particular interest is the Hopf algebra of the associahedra which has applications in quantum electrodynamics.

1.3 Further work

The cellohedra defined for finite regular cell complexes can likely be generalized to non-regular cell complexes where the boundary of an n-cell need not be homeomorphic to the (n − 1)-sphere, although the cellohedron for such a complex will likely fail to be bounded. Further the cellohedra seem to come close to describing what kind of sets can have a poset of tubings (special order ideals) that give a .

2 CHAPTER II

MOTIVATING CONCEPTS

2.1 The Classical Combinatorially-Described Polytopes

The preliminary theories and constructions presented in chapter two will motivate and clarify the later work of constructing cellohedra. The focus will be on the classical polytopes and their generalizations to graph and pseudograph associahedra.

2.1.1 The Associahedron

The associahedra were first constructed by Stasheff in his work studying H-spaces and associativity [8]. In his paper the associahedron of dimension n was defined as a cell complex whose cells correspond to legal bracketings of an n + 2 letter word, and no realization as a polytope was given. Since the associahedron’s resurfacing in the

field of operad theory, it has been given many realizations as a polytope. One of the simplest realizations is given by Jean-Louis Loday in terms of planar binary trees [5].

2.1.2 Planar Binary Trees and the Associahedron

A planar binary tree is a graph with no cycles that has exactly three edges incident to each internal node (the edges are designated: root, left branch, right branch). The set of all planar binary trees on n internal nodes (or n + 1 leaves) is denoted Yn.

3 We prefer to draw our trees opening upward, with the leaves all ending on the same horizontal line.

Enumerating the internal nodes from left to right using the set {1, 2, ..., n}, define ai to be the number of leaves to the left of and above node i, and define bi

n to be the number of leaves to the right and above node i. Define M : Yn −→ R

n to be the function that maps t ∈ Yn to the point (a1b1, a2b2, ..., anbn) ∈ R . The

n−1 associahedron K is defined to be the ConvexHull({M(t) | t ∈ Yn}). The map M

n−1 gives a from the elements of Yn to the vertices of K . Loday shows that the k-faces of Kn−1 are in bijection with planar trees with n + 1 leaves and n − k internal nodes (as defined in Stasheff’s original paper) [5]. Figures 2.1 and 2.2 show the points calculated via Loday’s algorithm and their convex hull.

2.1.3 The Permutohedron

Another example of a polytope related to a combinatorial structure is the permu- tohedron. The permutohedron is older than the associahedron and has a pleasant realization.

Let Sn be the set of all bijective functions from the set {1, 2, ..., n} to it-

n self. Define M : Sn −→ R to be the function that assigns σ ∈ Sn to the point

(σ(1), σ(2), ..., σ(n)) ∈ Rn. The permutohedron Pn−1 is defined to be ConvexHull({M(σ) |

σ ∈ Sn}).

4 (1, 2)

(2, 1)

(1,2,3)

(2,1,3) (1,4,1)

(3,1,2)

(3,2,1)

Figure 2.1: Points calculated for the associahedra in 1-dimension and 2-dimensions.

2.1.4 Relating the Permutohedron and the Associahedron

The elements of Sn can be described as planar binary trees with levels [9]. This provides a natural mapping φ : Sn −→ Yn obtained by ignoring the levels. Loday shows that this implies that Pn−1 ⊆ Kn−1 [5]. In Figure 2.3 we show two permuto- hedra, with leveled trees – note that there are just two leveled trees with the same underlying binary tree in 2d.

5 (1, 2)

(2, 1)

(1,2,3)

(2,1,3) (1,4,1)

(3,1,2)

(3,2,1)

Figure 2.2: The associahedra in 1-dimension and 2-dimensions.

2.2 Generalizing the Classical Polytopes

Let G be a graph with a set V and edge set E ⊆ V × V . Define a tube of G to be an induced graph of a proper of V , so a tube t is a proper connected subgraph of G that contains the edge (vi, vj) whenever it contains both vi and vj.

Two tubes t1 and t2 are nested if t1 ⊆ t2, intersecting if t1 ∩ t2 is nonempty, and adjacent if t1 ∩ t2 is a connected subgraph of G. Two tubes are compatible if they do not intersect and are not adjacent. A set of pairwise compatible tubes is called a

6 (1, 2)

(2, 1)

(1,2,3)

(2,1,3) (1,3,2)

(3,1,2) (2,3,1)

(3,2,1)

Figure 2.3: The permutohedra in 1-dimension and 2-dimensions.

tubing. A set of k pairwise compatible tubes is called a k-tubing.

Let ∆G be an n-dimensional with facets labelled with nodes of G.

The graph associahedron KG is defined to be the polytope obtained by truncating the faces of ∆G whose labeling corresponds to tubes of G (starting with vertices and increasing in dimension). The face poset of KG is isomorphic to the poset of tubings ordered by reverse inclusion.

7 The tubings of the path graph and the complete graph on three nodes are shown in Figure 2.4. Devadoss showed that the path graph associahedra are the classical associahedra, and the complete graph associahedra are the permutohedra.

( a ) ( b )

Figure 2.4: The associahedron and permutohedron in 2-dimensions.

The pseudograph associahedron is the extension of this to allow multiple edges and loops. A tube of a pseudograph G is defined to to be a proper connected subgraph of G that contains at least one edge (vi, vj) whenever it contains both nodes vi and vj. The definitions of adjacent, intersecting, nested, and compatible tubes are the same as above; as is the definition for a tubing. The pseudograph associahedron

KG is defined to be the polytope whose face poset is isomorphic to the poset of tubings ordered by inclusion. Figure 2.5 shows the tubings of a pseudograph which has two nodes and two edges.

8 Figure 2.5: The square as a pseudograph associahedron

The pseudograph associahedron KG can also be obtained by a truncation process, as shown in [2]. Let ∆k be a simplex with k-dimensional simplex, and let ρ be a ray. Define ∆G be the following polytope:

λ ∆G = ∆n−1 × Y ∆bi−1 × ρ (2.1)

Here n is the number of nodes in G, bi is the number of edges in the ith bundle of edges in G (all the edges of a bundle share the same pair of nodes), and λ is the number of loops in G. There is a labelling of faces of ∆G which we omit here, since it is re-described in a later section. Truncate ∆G along the faces corresponding to tubes of G which are induced subgraphs of G in increasing order based on the dimension of the face. Next truncate the rest of the faces which correspond to the remaining tubes in increasing order based on the number of nodes in the tube. Figures 2.6 and 2.7 show an example of the labeling of ∆G and the truncations that follow.

9 1 2 b 1 2 a

1 2 a

1 2 3 b 1 3 b 3 2 3 b a b 1 3 a 2 3 a

Figure 2.6: An example of a pseudograph and the labeling of ∆G.

1 a b 1 2 3 b 1 2 a b 1 2 a 1 2 b

1 2 3 a 3 a b 2 a b 2 3 a b

Figure 2.7: An example of a pseudograph associahedron obtained by truncation.

10 CHAPTER III

DEFINITIONS

3.1 Basic Topology of Rn

Let X and Y be of Rn. A bijection f : X → Y is a homeomorphism if and only if f and f −1 are continuous functions. X and Y are homeomorphic if and only if there exists a homeomorphism such that Dom(f)= X and Range(f)= Y . An open

n n n ball in R is a set B(r, x0) = {x ∈ R : |x − x0| < r}. An closed ball in R is a set

n n n B(r, x0)= {x ∈ R : |x − x0| ≤ r}. A sphere in R is a set {x ∈ R : |x − x0| = r}.

A set σ ∈ X is an open cell if and only if σ is homeomorphic to an open ball. Similarly a set σ ∈ X is a closed cell if and only if σ is homeomorphic to a closed ball. For σ ⊂ Rn, a point x ∈ σ is an interior point if and only if there exists an r > 0 such that B(r, x) ⊆ σ. The interior of σ is the set int(σ) = {x ∈ Rn : x is an interior point of σ}. A point x ∈ σ is a boundary point if and only if for every r > 0 the open ball B(r, x) ∩ σ and B(r, x) ∩ (Rn − σ) are both non empty. The boundary of σ is the set ∂σ = {x ∈ Rn : x is a boundary point of σ}.

11 3.2 Cell Complexes

Let X be a pairwise disjoint set of open cells in Rn. X is a cell complex if and only if for every σ ∈ X, ∂σ = S σk where each σk ∈ X. A cell complex is regular if and only if for every σ ∈ X, ∂σ is homeomorphic to an (n − 1)-sphere.

A cell complex X is finite if X contains a finite number of open cells. A cell complex is connected if and only if for every x, y ∈ X there exists a continuous function f : [0, 1] → X such that f(0) = x and f(1) = y. A set W is a subcomplex of

X if and only if W is a cell complex and W ⊂ X.

Let W be a connected subcomplex of X. W is a connected component of X if and only if for every connected subcomplex Y , if W is a subcomplex of Y then

Y = W . X can always be uniquely written as X1 ∪∪ Xk where each Xi is a connected component of X.

3.3 Tubes and Tubings

Let X = X1 ∪ ... ∪ Xk be a cell complex where X1,...,Xk are connected components of X. A tube t of X is a proper connected subcomplex of X such that for every open cell σ ∈ X if ∂σ ∈ t then the set Bσ = {τ ∈ X : ∂σ = ∂τ} ∩ t = ∅.

The set Bσ is called the bundle of σ. By convention tubes will be written as a list of cells with the brackets and commas omitted, for example {1, 2, a} would be written as 12a where 1,2, and a are cells of X. Two tubes are compatible if and

12 tubing

not a tube

not a tubing

Figure 3.1: An example of a tube and a non-tube.

only if one is a subcomplex of the other or they are disjoint and cannot be made to intersect by adding a 1-cell.

A tubing T of X is a set of pairwise compatible tubes not containing all of

X1,...,Xk. Figure 3.1 shows some examples and non-examples, where the second example is not a tube because it contains the bottom two nodes but no edge between them and the third example is not a tubing because the two tubes can be connected by a single 1-cell. The cellohedron KX is the poset of all tubings of X where the partial order ≺ is defined by T ≺ U if and only if T ⊃ U for T, U ∈KX.

The major thrust of this paper will be to argue that KX is isomorphic to the face poset of a convex polytope, and therefore the cellohedron for X, also called KX, is a unique convex polytope up to combinatorial equivalence.

13 CHAPTER IV

EXAMPLES

4.1 Some Three-Dimensional Examples

Example. Let X be the cell complex with two 2-cells (labeled A and B), two 1-cells

(labeled a and b), and two 0-cells (labeled 1 and 2); where ∂A = ∂B = a ∪ b ∪ 1 ∪ 2 and ∂a = ∂b = 1 ∪ 2. The tubes of X are: 1, 2, 12a, 12b, 12abA, 12abB. Each tube is compatible with every tube that has a different number of cells in it, and KX is the

3-dimensional cube.

a a

1A 2 1 2

b a 2

1AB 2

b 1 a B 1 2 1 2

b b

Figure 4.1: The 3-dimensional cube as a cellohedron

14 Example. Let X be the cell complex with two 2-cells (labeled A and B), three 1- cells (labeled a,b, and c), and two 0-cells (labeled 1,2, and 3); where ∂A = ∂B = a ∪ b ∪ c ∪ 1 ∪ 2 ∪ 3 and ∂a =1 ∪ 2, ∂b =2 ∪ 3, and ∂c =1 ∪ 3. The tubes of X are:

1, 2, 3, 12a, 23b, 13c, 123abA, 123abB. The tubes 123abcA and 123abcB are compatible with every tube but each other, and the rest of the tubes are compatible with only two other tubes. KX is the hexagonal prism.

2

2 a b A a B b 1 3 A c 2 2 1 3 c b 2 a 3

1

1 3 1 3 2 c

a B b

1 3 c

Figure 4.2: The hexagonal prism as a cellohedron

15 4.2 A Four-Dimensional Example

Example. Let X be the cell complex from the Hexagonal Prism example with an extra 1-cell labeled d and an extra 0-cell labeled 4, with ∂d = 3 ∪ 4. The tubes of X are:

1, 2, 3, 4,

12a, 23b, 13c, 34d,

134cd, 234bcd, 123abcA, 123abcB,

123abcAB, 1234abcdA, 1234abcdB.

The cellohedron KX is drawn as a Schlegel diagram in Figure 4.3. Figures 4.4 and 4.5 show the 15 facets of the cellohedron with their corresponding tubes.

4.3 Some Easy Cases

In this section two proofs are given, which describe the cellohedra related to certain

”nice” sequences of cell complexes.

4.3.1 A Cycle Filled in with Sheets

Let X be the cell complex constructed as follows:

1. Let X0 = {1,...,n}

2. Attach the set of 1-cells {e1,...,en} where ∂ei is attached to i and i +1 for

1 ≤ i ≤ n − 1, and ∂en is attached to n and 1. The resulting cell complex is

X1.

16 3. Attach the set of 2-cells {d1,...,dk} where each ∂dj is attached to S ei. The

resulting cell complex is X2 = X.

Let U ∈ KX be a tubing such that U does not have any tubes that contain 2-cells.

Since X1 is a cycle graph and U cannot contain a tube containing every 0-cell, U is equivalent to a tubing in KCn where Cn is the cycle graph on n nodes. Let α(U) be this equivalent tubing. Let V ∈ KX be a tubing that has only tubes that contain

2-cells. A tube in V can be described as a subset of {d1,...,dk} and compatibility of these tubes is equivalent to the subsets being nested.

Let Γk be the compete graph with vertices labeled {v1,...,vk}. Let β(U) be

the tubing on Γk such that if {dj1 ,dj2 ,...,djl } is a tube in U, {vj1 , vj2 ,...,vjl } is a tube in β(U). Any tubing T ∈KX can be written as a tubing U and a tubing V .

Let ϕ(T )=(α(U), β(V )). It is clear, from the previous description of α and β, that this map preserves compatibility, and is bijective. Thus KX is isomorphic to

KCn ×KΓk = Wn × Pk. Figure 4.6 shows the 3d (KC4) crossed with the

1d permutohedron.

4.3.2 Double Bundles of Every Dimension

∞ Let {Xn}n=0 be a sequence of cell complexes where X0 is two disconnected 0-cells labeled {1, 2}, and Xn is the cell complex obtained by adding two n-cells with their boundaries attached to all of Xn−1. The cellohedron for X0, X1 and X2 have been shown to be an interval, a square, and a cube respectively.

17 Suppose it is true that KXk−1 is a k-dimensional for some k ≥ 1, and let T ∈ KXk. Let U be the set of tubes in T that do not include either of the k-cells in KXk, and let V be the set of tubes in T that do contain a k-cell. The set

U is a tubing on Xk−1, and define α to be the map that takes the set U to a tubing on KXk−1. Since the there are only two tubes of Xk containing k-cells (call them A and B), and these tubes are not compatible, V contains only one or the other.

Let G0 be a graph with two vertices labeled {vA, vB} with a single edge between them, and define β(V ) to be the tubing in KG0 that contains only the tube with vA when V = {A} and only the tube with vB when V = {B}. These two tubings are all of KG0, and the only possible sets V for a tubing T . Since any tube

T can be divided uniquely this way, the map ϕ(T )=(α(U), β(V )) is an isomorphism between KXk and KXk−1 ×KG0. Since KXk−1 is a hypercube and KG0 is a line segment, KXk becomes the (k + 1)-dimensional hypercube. By induction KXn is the

(n + 1)-dimensional hypercube for n ≥ 0.

18 2 4

a B b d A

1 3 c

Figure 4.3: A 4-dimensional cellohedron

19 3

2 4

a b d A 2 1 3 c

1

4

Figure 4.4: Facets from Figure 4.3.

20 2

2 a b 2 A a b 1 3 c a B b

1 3 c 1 3 c

4 2 4

d b d

1 3 c 3 2 4

a B b d

1 3 c

2

b

3

1 3 2 c a

1 4

d

3

Figure 4.5: More facets from Figure 4.3.

21 Figure 4.6: The cellohedron of a four cycle filled in with two sheets

22 CHAPTER V

THEOREMS

5.1 Single Bundles Have a Trivial Effect on KX

Theorem 1. Let X be a cell complex and W = {σ ∈ X : dim(σ) > 1, |Bσ| =

1, and σ ∩ ∂τ = ∅for all τ ∈ X}. If X∗ = X − W , then KX is isomorphic to KX∗.

Proof. Define ϕ : tubes of X → tubes of X∗ by ϕ(t) = t − W . Since X is regular and no element of W is part of the boundary of any other cell in X, ϕ(t) is a tube of X∗ for every tube t. Since 1-cells and nesting are not affected by removing W , compatibility is preserved by ϕ. Finally since a tube containing a cell in W is forced to by the condition Bσ ∩ t > 0 whenever ∂σ ∈ t, the tube ϕ(t) is not already a

∗ tube of X. Therefore the map Φ : KX → KX defined by Φ(T = {t1,...,tk}) =

{ϕ(t1),...,ϕ(tk)} is a poset isomorphism giving the above result.

5.2 KX Can Be Expressed In Terms of Connected Components and the Edgeless

Graph

Theorem 2. If X = X1 ∪ ... ∪ Xk where each Xi is a connected component of X, then KX is isomorphic to KX1 × ... ×KXk × ∆k−1.

23 Proof. Let T ∈ KX. T can be partitioned into subsets T1,...,Tk, and U where

Ti ∈ KXi and U ⊂ {X1,...,Xk} which is isomorphic to the edgeless graph. Since each subset can be selected independently from the other subsets, the map Φ(T ) =

(T1,...,Tk, U) ∈KX1 × ... ×KXk × ∆k−1 is an isomorphism.

5.3 The Truncation Method

For each cell σ ∈ X assign a label lσ. Given a cell complex X, define △△X by:

△△X = ∆|X(0)|−1 × Y ∆|Bσ|−1 (5.1)

where the product runs over each bundle of X. Label the facets of ∆|X(0)|−1 with the

labels of each 0-cell of X, and label the vertices of ∆|Bσ|−1 with the labels of each element of Bσ. The labeling of △△X is induced by the labeling of these simplices. A tube t is labeled by lt where if σ ∈ t then lσ ∈ lt; and if Bσ ∩ t is empty, then lτ ∈ t

∗ for each τ ∈ Bσ. Define △△X to be the polytope obtained by truncating the faces of

△△X corresponding to full tubes in increasing order of dimension.

∗ (1) Theorem 3. △△X = K(X )S)×Q ∆|Bσ|−1, where the product runs over every bundle of X.

Proof. Let t be a full tube of X. Since t either contains all of a bundle or none of the bundle, the label of t must include lσ for every cell σ ∈ X with dim(σ) > 0. The faces of △△X that contain all of these lσ’s is a product of a face of ∆|X(0)|−1 and the

entire simplex ∆|Bσ|−1 for each bundle Bσ. Truncating these faces has a trivial effect

24 1 2 b 1 2 a

1 2 a

1 2 3 b 1 3 b 3 2 3 b a b 1 3 a 2 3 a

Figure 5.1: An example of labeling a △△X

on the second part of this product, and truncating the faces of the first part gives

(0) K(X )S.

∗ Theorem 4. Truncating the remaining faces of △△X that correspond with tubes of

X gives the cellohedron KX, and therefore KX is a convex polytope with dimension n − 1+ r where r = (|B | − 1). PBσ σ

The proof of this theorem will be omitted here it will be completed for the published version. The following figures show how the 4d example shown in Figure 4.3 can be obtained by truncations.

25 123A

2 4 234A 123B 234B a B b d A 124B 134B 1 3 c

134A 124A

Figure 5.2: Vertices are labeled. These labels extend to faces of all dimensions. 123AB 234AB

134AB

Figure 5.3: These three labeled edges correspond to full tubes.

26 Figure 5.4: The preceding three labeled edges have been truncated.

Figure 5.5: Now we have truncated the four 2-faces corresponding to non-full tubes with only two nodes and one edge.

27 Figure 5.6: Finally we have truncated the two 2-faces corresponding to non-full tubes with three nodes.

28 BIBLIOGRAPHY

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[2] M. Carr, S. Devadoss and S. Forcey. Pseudograph associahedra, Journal of Combinatorial Theory, Series A 118 (2011) 2035–2055.

[3] S. Devadoss. A realization of graph associahedra, Discrete Mathematics 309 (2009) 271–276.

[4] S. Forcey and D. Springfield. Geometric combinatorial algebras: cyclohedron and simplex, Journal of Algebraic 32 (2010) 597–627.

[5] J.-L. Loday. Realization of the Stasheff polytope, Archiv der Mathematik 83 (2004) 267–278.

[6] A. Postnikov. Permutohedra, associahedra, and beyond, International Mathe- matics Research Notices 6 (2009) 1026–1106.

[7] A. Postnikov, V. Reiner, L. Williams. Faces of generalized permutohedra, Doc- umenta Mathematica 13 (2008) 207–273.

[8] J. D. Stasheff. Homotopy associativity of H-spaces I. Trans. A. M. S. 108 (1963), 275-292.

[9] A. Tonks. Relating the associahedron and the permutohedron, in Operads: Pro- ceedings of Renaissance Conference, Contemporary Mathematics 202 (1997) 33–36.

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29