THREE CONFIGURATION SPACES IN COMBINATORICS

A thesis presented to the faculty of San Francisco State University A ^ In partial fulfilment of The Requirements for The Degree

• JoR

Master of Arts * (? In Mathematics

by

John Guo

San Francisco, California

August 2017 Copyright by John Guo 2017 CERTIFICATION OF APPROVAL

I certify that I have read THREE CONFIGURATION SPACES IN

COMBINATORICS by John Guo and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at

San Francisco State University.

Jldmbx QadJe/fy. Federico Ardila Professor of Mathematics

Serkan Hosten Professor of Mathematics

Dustin Ross Professor of Mathematics THREE CONFIGURATION SPACES IN COMBINATORICS

John Guo San Francisco State University 2017

Roughly speaking, a reconfigurable system TZ is a discrete collection of positions of an object, along with local reversible moves. Such a system can be encoded as a cubical complex, which we call the configuration space S{1Z). When a configuration space is CAT(O), there is a unique shortest path between the vertices, and there is an efficient algorithm to compute this path. We can assess whether or not a cubical complex is CAT(O) by determining if there exists a corresponding poset with inconsistent pairs (PIP). In this thesis we show that the configuration spaces of the robotic arm in a rectangular tunnel Rm,n, of the tableaux T\ of hook shape

A, and of the Dyck paths Dn of length 2n are CAT(O) cubical complexes. We do this by identifying the associated PIPs.

I certify that the Abstract is a correct representation of the content of this thesis. $kbdwQM&°/l/y______M n w * Chair, Thesis Committee Date ACKNOWLEDGMENTS

The content of Chapter 4 on the robotic arm Rm,n is joint work with

Federico Ardila, Hanner Bastidas, and Cesar Ceballos [3]. The material on Knuth equivalence in Chapter 5 is collaborative work with Anastasia

Chavez.

I would like to especially thank Federico for his enthusiasm and guidance.

He has been just as much a friend, as an advisor.

I want to thank Anna Schindler and Andres Vindas most of all, for being there during both the fun and the hard parts. Thank you for making sense of my rambling.

v TABLE OF CONTENTS

1 Introduction...... 1

1.1 The Robotic Arm Rm,n ...... 1

1.2 Knuth Relations and Tableaux T\ ...... 3

1.3 Dyck Paths D n ...... 4

1.4 Methods...... 5

1.5 Organization ...... 6

2 Reconfigurable Systems ...... 8

3 CAT(O) Cubical Complexes and PIPs ...... 12

4 The Robotic Arm in a Tunnel ...... 17

4.1 Face Enumeration and the Euler Characteristic of

4.1.1 Cubes and partial s ta te s ...... 19

4.1.2 Factorization of partial states into irreducibles ...... 21

4.1.3 Enumeration of irreducible partial s ta te s ...... 23

4.1.4 Enumeration of irreducible final partial s t a t e s ...... 25

4.1.5 The /-vector and Euler characteristic of S(R,2,n) ...... 27

4.2 Coral P I P ...... 28

4.3 Enumeration of Tight States in

5 Tableaux and Knuth E quivalence...... 44

vi 5.1 The Insertion and Recording Ta b le a u x ...... 46

5.2 The Bump PIP B\ ...... 49

6 Dyck Paths ...... 55

6.1 Face Enumeration and the Euler Characteristic of S (D n) ...... 56

6.2 Valley P I P ...... 62

Bibliography ...... 69

vii LIST OF TABLES

Table Page

4.1 The /-vectors of the cubical complexes

n < 6...... 19

4.2 The eight types of irreducible partial states and their generating func­

tions...... 24

4.3 The eight types of irreducible final partial states and their generating

functions...... 26

4.4 The number of states and tight states, with asymptotics. Both c and

d are constants...... 43

5.1 The RS algorithm of n = 87561234...... 47

5.2 The elements in the order ideal of {B 2,3(1), B5fi(3 )}...... 54

viii LIST OF FIGURES

Figure Page

1.1 A position of the robotic arm # 3,14...... 2

1.2 The two types of local moves of Rm,n, switching corners and flipping

the end...... 2

1.3 The Knuth equivalence classes of S4...... 4

1.4 The coral PIP 62,9, bump PIP J5(5;13), and the valley PIP U7 ...... 7

3.1 A chord in X and the corresponding chord in R2...... 13

3.2 A PIP and the corresponding rooted CAT(O) cubical complex. . . . 14

4.1 The configuration space

tunnel of width 2. The horizontal arm corresponds to the bottom

most vertex...... 18

4.2 A partial state corresponding to a 6-cube in the configuration space

S(R 2,20) ...... 20

4.3 The partial state of Figure 4.2 has a factorization of the form M 1M 5M 1F7 .

(See Tables 4.2 and 4 . 3 . ) ...... 21

4.4 Illustration of condition (ii) in Definition 4.10...... 29

4.5 A mathematical coral snake and (a photograph of) a real-life coral

snake...... 30

4.6 Two coral tableaux; the one on the right is tight...... 31

4.7 From a state R to a coral tableau T ...... 32

4.8 A tight state and its corresponding tight tableau...... 32 4.9 The PIP Cm)n for m = 2 and n — 9...... 35

4.10 A coral tableau T = Ti V • • • V T6 = Tx V T2 V T4 V Te as a join of

irreducibles...... 37

4.11 The cube corresponding to the moves above...... 38

4.12 The 10 tight states of i?2,4...... 40

4.13 The factorization of a tight state of i?2,i7- Excluding the first three

steps, this is the factorization of Proposition 4.16...... 41

5.1 The configuration space S(Tu68i257)...... 45

5.2 A Young diagram of shape A = (3, 2,2,1) and a standard Young

tableau of shape A...... 46

5.3 The configuration space iS(T(4j4)), also expressed in terms of tableaux. 48

5.4 The cubical complex of the canonical equivalence class of shape A =

(5, l 3), with the corresponding bump PIP...... 51

6.1 A Dyck path of D 7 and the local move...... 55

6.2 A partial state corresponding to a 4-cube in S(D±2)...... 57

6.3 The decomposition of Figure 6.2 into square-irreducible partial states. 58

6.4 How three irreducible states concatenate as a square-irreducible. . . . 60

6.5 The first five configuration spaces of D n. The upright and floor paths

are emphasized...... 63

6.6 A valley Dyck path in Dj and the valley PIP Uj...... 66

x 1

Chapter 1

Introduction

Loosely speaking, a reconfigurable system is a collection of different positions of an object, along with reversible local moves that navigate between the positions.

We consider three reconfigurable systems: the robotic arm Bm/n of length n in a rectangular tunnel of width m, of the standard Young tableaux T\ of shape A where

A is a partition of n, and the Dyck paths Dn of length 2n.

1.1 The Robotic Arm R m^n

The robotic arm of length n in a tunnel of width m consists of n sequential unit length steps restricted to the grid. The initial step begins at the lower left corner of the grid, and each subsequent step must travel up, down, or right. The arm must never self-intersect or leave the grid. Figure 1.1 shows a position of the robotic arm of length 14 in a tunnel of width 3.

We begin with the fully horizontal arm and consider all the possible positions 2

Figure 1.1: A position of the robotic arm # 3,14.

of the arm that can be attained using two types of local moves: switching corners

and flipping the end. Switching corners takes two consecutive segments which form

a corner and replaces them with the two segments forming the opposite corner.

Flipping the tail rotates the final segment by 90°. These are illustrated in Figure

1.2. It is crucial that the robotic arm stays on the grid and never self-intersects.

□ — n 1 • - h

Figure 1.2: The two types of local moves of Rm^n, switching corners and flipping the end.

A question one can hsk is to determine a shortest sequence of moves between

any two positions of the robotic arm. This question clearer in the context of the

configuration space

sitions of the arm and whose d-cubes correspond to d-tuples of commutative moves.

Ardila, Baker, and Yatchak [2] showed the question can be answered efficiently if the configuration space is a CAT(O) cubical complex. This motivates one of our three main theorems. 3

Theorem 1.1. The configuration space S (R mtn) of the robotic arm Rm,n in a tunnel is a CAT(O) cubical complex.

Corollary 1.2. [1 , 2, 4] There is an explicit algorithm to move the robotic arm

Rm,n optimally from one state to another.

The link http://math.sfsu.edu/federico/Articles/movingrobots.html leads to Python code containing this algorithm for the robotic arm Rm,n, along with a graphic.

1.2 Knuth Relations and Tableaux T\

Next we consider a permutation n in the symmetric group Sn. We write n in one-line notation, that is, it — 7Ti7r2 ... 7rn where = it(i). A well-known equivalence relation in the context of representation theory is Knuth equivalence. When x < y < z are consecutive entries of the permutation n, there are two possible Knuth relations: yxz = yzx and xzy = zxy. Two permutations n and a are said to differ by a Knuth relation if the only difference between them is given by a single Knuth relation. We 1 2 say 7T = a if the difference is given by yxz = y z x , and n = a if the difference is a

K result of xzy = zxy. Moreover ir and a are Knuth equivalent, or tt = cr, if there is a sequence of Knuth related permutations between them.

If 7r is a permutation of Sn, we can define a second reconfigurable system Tn.

K The states of Tn are all the permutations a such that a = n, and the moves change 4

1243 “ 1423 “ 4123 3124 S 1324 ^ 1342 1432 “ 4132 ^ 4312 4213 S 4231 “ 2431 2134 S 2314 ^ 2341 3214 “ 3241 3421 3142 “ 3412 2143 i 2413 1234 4321

Figure 1.3: The Knuth equivalence classes of S between the Knuth related permutations of Tn. It turns out that these classes of permutations are more generally described by tableaux given by the Robinson-

Schensted algorithm discussed in Section 5.1. We pay special attention to those

Knuth equivalence classes which correspond to tableaux that are canonically filled, that is tableaux whose entries increase by one from left to right, top to bottom. We refer to such reconfigurable systems as T\, where A is the shape of the canonically filled tableaux. We can again ask whether or not the configuration space S(T\) is a CAT(O) cubical complex. We prove that this is only the case when A is a hook, that is, when A consists of one row and one column.

Theorem 1.3. The configuration space S(T\) of canonically filled tableaux of shape

X, is a CAT{0) cubical complex if and only if the shape X is a hook.

1.3 Dyck Paths D n

A Dyck path of length 2n is similar to a robotic arm. The path travels from (0,0) to (2n,0) using two types of steps: an upstep (1,1), and a downstep (1 ,-1 ). No segment can fall below the horizontal axis. We can define a move to be a change 5

between a peak and a valley, or in other words interchanging an adjacent downstep

and upstep. We refer to the collection of all Dyck paths of length 2n as Dn. We show the corresponding configuration space S (D n) is also a CAT(O) cubical complex.

Theorem 1.4. The configuration space S (D n) of the Dyck paths of length 2n is a

CAT(0) cubical complex.

1.4 Methods

A metric space is CAT(O) when the space is without positive curvature [6]. A space

has nonpositive curvature if every possible geodesic triangle is “thin” , in a sense

to be defined later. These spaces have been of large interest to geometric group

theorists. We are not aware of many infinite families of CAT(O) cubical complexes,

which is why we provide three more. In the case of the robotic arm, it was shown

that in a tunnel of width 1, the space S(Ri,n) is CAT(O) [2], We generalize this

result to a tunnel of arbitrary width.

It is not clear how to show a space is CAT(O) using the geodesic triangles def­

inition. For cubical complexes, Gromov gave a different topological approach to

showing when they are CAT(O): a finite-dimensional cubical complex is CAT(O)

if and only if it is simply connected and the link of every vertex is a flag simpli-

cial complex [9]. In the context of reconfigurable systems, the associated cubical

complex automatically satisfies the link condition [8]. Ardila, Owen, and Sullivant 6

[4] give an alternative combinatorial approach to showing whether or not a cubical complex is CAT(O), of which we make heavy use. This combinatorial approach de­ pends on a bijection between CAT(O) complexes with a specified vertex, or root, and posets with inconsistent pairs (PIPs). Thus to know if the configuration space of a reconfigurable system is CAT(O), it is enough to identify a matching PIP.

1.5 Organization

In Chapter 2, we build up the basic concepts of reconfigurable systems, and most importantly note that each reconfigurable system can be completely described by a cubical complex. In Chapter 3 we define what it means for a space to be CAT(O) and elaborate on the bijection between rooted CAT(O) cubical complexes and PIPs.

Each of the following sections is dedicated to a reconfigurable system defined above. Chapter 4 is devoted to the robotic arm. We redevelop the notion of the configuration space of the robotic arm SiRm^) and, using generating functions, prove has Euler characteristic 1. This is a suggestive result since CAT(O) spaces are known to be contractible and hence have Euler characteristic equal 1.

Then we define the necessary tools to prove that the coral PIP Cm>n is the PIP of the robotic arm, and conclude the topic of the robotic arm by enumerating the elements of the coral PIP when m — 2.

In Chapter 5 we discuss the reconfigurable system given by Knuth equivalence

Tn. We briefly review the Robinson-Schensted algorithm relating permutations and 7

tableaux. This brings us to the broader reconfigurable system of tableaux, T\. We show that the configuration space S(T\) is a CAT(O) cubical complex when A is a hook by identifying it with the bump PIP B\. It should be pointed out that the

1-skeleton, consisting solely of the edges and vertices of S(T\), has been studied under the slightly different name of dual equivalence graphs [5].

Figure 1.4: The coral PIP C^g, bump PIP and the valley PIP U7 .

We cover Dn, the Dyck paths of length 2n, in Chapter 6. We begin this chapter by constructing the generating function of the face vector for S(D n) and show the

Euler characteristic is 1. We show this reconfigurable system also produces a CAT(O) cubical complex by identifying it with the valley PIP Un. Chapter 2

Reconfigurable Systems

Defining reconfigurable systems rigorously feels distant from the intuition. We in­

clude this section for familiarity of the terms and to also unite our three main objects, which are related to each other fundamentally as reconfigurable systems.

The definitions in this section make the most sense in terms of an example, but because we have three reconfigurable systems, we introduce the definitions in full generality. The reader may find it useful to read these definitions with an example in mind, such as the ones in Chapter 4, 5, 6.

Introduced by Abrams, Ghrist, and Peterson [1, 8], a reconfigurable system

consists of a graph G, a collection of labelings of the vertices of G known as states,

and generators which locally alter these vertex labelings. Generators understand

the legality of altering a labeling, known as the support, and generators know where

this change takes place; this is referred to as the trace.

Let A be a set of labels, and G be a graph. 9

Definition 2.1. A state is a labeling of each vertex of G by an element of A.

Definition 2.2. A generator consists of the three objects:

1. A subgraph SUP (cf>) C G, known as the support of

2. A subgraph TR() C SUP(0), known as the trace of 4>.

3. Let V(G) by the vertices of the graph G. An unordered pair of local states

ul0°c,u[°c : V ( S U P (0 ))^ A ,

which are labelings of the vertices of SUP() by elements of A. These local

states must agree on SUP(^) — TR(). In other words the trace is the move,

or the difference of labelings.

We call a generator (f> admissible at a state u if w|sup() £ {«o°c,«]°c}. If 4> is admissible, the action of 4> on state u is denoted as 4>[u}. The action interchanges between the labelings Uqoc and ul°° on the support and otherwise leaves the state u alone. This results in a new state as follows

u : on G - SUP() { : on SUP(<£), where u|sup(<« = ul°c Moreover these moves are reversible in the following sense, [[«]] = u because the pair of local states of is unordered. 10

More directly, when a state u is labeled in the same way as one of the local states of a generator (f> on the vertices of the support of , then is an available move.

What (f) does is toggle between the two different labelings of the support, u1q° and ul°c, and the difference between these two local states is precisely on the trace of , leaving u labeled the same elsewhere.

Definition 2.3. A reconftgurable system on G is a collection of generators and a collection of states closed under all possible admissible actions.

An important relationship between two moves is whether or not they are inde­ pendent of each other. If two generators fa and fa affect far enough subgraphs of

G, then neither generator inhibits the other move from being viable. We call such moves physically independent or commutative, and if both fa and fa are admissible at state u, it does not matter the order in which we apply them.

Definition 2.4. A collection of admissible generators {fa : i E 1} is said to commute if

TR(fa) n SUP(^) = 0 for all i + j in /.

For the purpose of this thesis, an important feature of a reconfigurable system is that it can be encoded as a cubical complex. We call this encoding the configuration space.

Definition 2.5. A cubical complex A is a polyhedral complex obtained by gluing together cubes of various dimensions such that the intersection of any two cubes is 11

a face of both.

Definition 2.6. The configuration space S(TZ) of a reconfigurable system 7Z is the following cubical complex. The vertices correspond to the states of 7Z. There is an edge between vertices u and v if there is an admissible move which takes the reconfigurable system between u and v. The k-cubes correspond to fc-tuples of commutative moves: with k moves applicable at a state u, we can obtain 2 k distinct states from u by performing a subset of these k moves; these are the vertices of a fc-cube in S(JZ). Defining each /e-cube to be a unit cube provides S(7Z) with a

Euclidean metric. 12

Chapter 3

CAT(O) Cubical Complexes and PIPs

To define a CAT(O) space we require a metric. Let X be a metric space such that every pair of points x ,y has a unique shortest path with length d(x,y), called a geodesic. Consider two triangles, a triangle T with geodesic sides of lengths a, b, c in

X, and a triangle T' with identical side lengths in the Euclidean plane. For any two points on the boundary of T we can consider the geodesic chord between them of length d. On T' there are two analogous boundary points, and suppose the geodesic chord between them has length d!. We call the triangle T thin if every geodesic chord in T is shorter than the analogous chord in T', that is d < d' for all chords in

T. Moreover, X has non-positive global curvature if every triangle in X is thin. See

Figure 3.1.

Definition 3.1. A metric space X is CAT(O) if there is a unique geodesic between any two points in X, and X has non-positive global curvature.

The first major approach to demonstrating when a cubical complex is CAT(O) 13

Figure 3.1: A chord in X and the corresponding chord in R 2. was given by Gromov; for this we must define a few terms. If A is a cubical complex, the link of a vertex v is the abstract simplicial complex A whose fc-dimensional simplices are the (k + l)-dimensional cells incident to v with the natural boundary relationships [8]. The simplicial complex A is flag if for every k + 1 vertices of A, where each pair is joined by an edge, the corresponding /c-sirnplex is also in A [9].

The space X is simply connected if it is path connected, and any path starting and ending at the same point xo, can be continuously transformed into the trivial path which remains at the point xq [12],

Theorem 3.2. [9] A cubical complex is CAT(O) if and only if it is simply connected and the link of every vertex is a flag simplicial complex.

Ardila, Owen, and Sullivant gave another method by showing finite dimensional rooted CAT(O) cubical complexes are in bijection with posets with inconsistent pairs

(PIP) [4]. This bijection generalizes that of Birkhoff’s representation theorem, which relates distributive lattices to posets.

D efinition 3.3. A poset with inconsistent pairs (PIP) is a poset P together with a 14

collection of inconsistent pairs, which we denote p q (where p ^ q), such that

1. If p and q are inconsistent, then there is no r such that r > p and r > q.

2. If p and q are inconsistent and p' > p, q' > q, then p' and q' are inconsistent.

Similar to posets, a PIP can be represented graphically by a Hasse diagram.

Property 2 says that order preserves inconsistencies, thus in the Hasse diagram of a

PIP it is enough to denote minimal inconsistent pairs. The Hasse diagram of a PIP

is the Hasse diagram of the poset with dotted lines between minimal inconsistent

pairs. Notice also that two inconsistent elements must also be incomparable.

Figure 3.2: A PIP and the corresponding rooted CAT(O) cubical complex.

Theorem 3.4. [\] There is a bijection P (->• X(P) between posets with inconsistent

pairs and rooted CAT(O) cubical complexes. 15

With this bijection, if we wish to show that a cubical complex is CAT(O), it suffices to pick a vertex as our root and determine the corresponding PIP. Let us elaborate on this bijection as it is the driving force for the results in this thesis.

Rooted CAT(O) cubical complex t— » PIP: A CAT(O) cube complex X has a system of hyperplanes as described by Sageev [14] . Each d-cube C in X has d “hyperplanes” of codimension 1, each one of which is a (d— l)-cube orthogonal to an edge direction of C and splits it into two equal parts. When two cubes C and C' share an edge e, we identify the two hyperplanes in C and in C' that are orthogonal to e. The result of all these identifications is the set of hyperplanes of X. Each hyperplane divides X into two disjoint pieces. The right panel of Figure 3.2 shows a CAT(O) cube complex and its six hyperplanes.

Fix v. Now we define the PIP P associated to (X,v). The elements of P are the hyperplanes of X. For hyperplanes Hi and H2, we declare Hx < H2 if, starting at the root v of X, one must cross hyperplane H\ before one can cross hyperplane

H2. Finally, we declare Hi H2 if, starting at the root v of A , it is impossible to cross both Hi and H2 without backtracking. The left panel of Figure 3.2 shows the

PIP associated to the rooted complex of the right panel.

PIP i— > rooted CAT(O) cubical complex: Let P be a PIP. An order ideal of P is a subset / such that if x < y and y G I then x G I. We say that I is consistent if it contains no inconsistent pair.

The vertices of X (P ) are identified with the consistent order ideals of P. There 16

is a cube C(I,M) for each pair (I,M) of a consistent order ideal / and a subset

M C Imaxi where Imax is the set of maximal elements of I. This cube has dimension

\M\, and its vertices are obtained by removing from I the 2^MI possible subsets of

M. The cubes are naturally glued along their faces according to their labels. The root is the vertex corresponding to the empty order ideal.

This bijection is also illustrated in Figure 3.2; the labels of the cubical complex on the right correspond to the consistent order ideals of the PIP on the left. Note that for a CAT(O) cubical complex X(P), different choices of roots can give very different PIPs. For the three configuration spaces S(Rmtn),S(T\), and S(Dn), our choice of root is carefully chosen to minimize inconsistent pairs. 17

Chapter 4

The Robotic Arm in a Tunnel

Recall that the robotic arm Rm,n of length re in a tunnel of width m consists of n sequential unit length steps restricted to the grid. The initial step begins at the lower left corner of the grid, and each subsequent step must travel up, down, or right. The arm must never self-intersect or leave the grid.

The two types of available moves are switching comers and flipping the end.

Switching corners takes a connected vertical and horizontal segment which form a corner and moves them to form the opposite corner. Flipping the end takes the last segment and rotates it 90°. These two moves are shown in Figure 1.2. A move is only available when it does not cause the arm to intersect itself or exit the tunnel.

The reconfigurable system Rm,n is the collection of all possible positions, or states, of the robotic arm which can be obtained by starting from the completely horizontal arm and applying these two types of moves. As stated in Chapter 2, Rm,n can be encoded as a cubical complex. 18

w

Figure 4.1: The configuration space 5 (i?2,e) of the robotic arm of length 6 in a tunnel of width 2. The horizontal arm corresponds to the bottom most vertex.

Definition 4.1. The configuration space S (R mtn) of the robot Rm,n is the following cubical complex. The vertices correspond to the states of Rm,n■ There is an edge between vertices u and v if there is a legal move which interchanges between u and v. The fc-cubes correspond to /c-tuples of commutative moves.

4.1 Face Enumeration and the Euler Characteristic of S (i?2,n)

One of the main structural results of this thesis, Theorem 1.1, is that the config­ uration space S(Rm,n) of our robot is a CAT(O) cubical complex. This is a subtle metric property defined in Chapter 3 and proved in Section 4.2. As a prelude, this section is devoted to proving a partial result in that direction. It is known [6, 9] that CAT(O) spaces are contractible, and hence have Euler characteristic equal to

1. We now prove:

Theorem 4.2. The Euler characteristic of the space

This provides enumerative evidence for our main result in width m = 2. While we were stuck for several weeks trying to prove Theorem 1.1, we found this evidence very encouraging.

To prove Theorem 4.2, our strategy is to compute the /-vector of S (R 2tn). Recall that the f -vector of a d-dimensional polyhedral complex X is f x = {fo, fi, ■ • •, fd) where /& is the number of /c-dimensional faces. The Euler characteristic of X is x(X) = fo — fi + • • • + (—1 )dfd- Table 4.1 shows the /-vectors of the cubical complexes of the robotic arms of length n < 6. For instance, the complex of Figure

4.1 contains 53 vertices, 81 edges, 30 squares, and 1 cube.

n fo h h f 3 X(S(R2,n)) 1 2 1 0 0 1 2 4 3 0 0 1 3 8 8 1 0 1 4 15 18 4 0 1 5 28 38 11 0 1 6 53 81 30 1 1

Table 4.1: The /-vectors of the cubical complexes S{R^,n) for arms of length n < 6.

We compute the generating function for the /-vectors of the configuration spaces

[16] or [17] for details. We proceed in several steps.

4.1.1 Cubes and partial states

Consider a d-cube in the configuration space iS(i?2,«); it has 2 d vertices. If we superimpose the corresponding 2d positions of the robotic arm, we obtain a sequence 20

of edges, squares, and possibly a “claw” in the last position, as illustrated in Figure

4.2. The number of squares (including the claw if it is present) is d, corresponding to the d physically independent moves being represented by this cube. We call the resulting diagram a partial state, and let its weight be x nyd. The partial states of weight xnyd are in bijection with the rl-cubes of

Figure 4.2: A partial state corresponding to a 6-cube in the configuration space <5(^2,2o)-

Each partial state gives rise to a word in the alphabet {r, v, □ , l } , where:

• r represents a horizontal step of the robot facing to the right. Its weight is x.

• v represents a vertical step. Its weight is x.

• □ represents a square, which comes from a move that switches corners of two consecutive links facing different directions. Its weight is x 2y.

□ l - - ~ i

• l represents a claw, which comes from a move that flips the end of the robot, with the horizontal step facing to the right. Its weight is xy.

t I 21

For example, the partial state of Figure 4.2 gives rise to the word

w = rnnruDrDrrnrruL.

The weight of the partial state is the product of the weights of the individual sym­ bols; in this case it is

x (x 2y)(x2y)xx(x2y)x(x2y)xx(x2y)xxx(xy) = x 20y6.

It is worth remarking that this word does not determine the partial state uniquely.

4.1.2 Factorization of partial states into irreducibles

Our next goal is to enumerate all partial states according to their length and di­ mension. The key idea is to “factor” a partial state uniquely as a concatenation of irreducible factors. Each time the partial state arrives to the top or bottom border of the tunnel, we start a new factor. For example, the factorization of the partial state of Figure 4.2 is shown in Figure 4.3.

Figure 4.3: The partial state of Figure 4.2 has a factorization of the form M 1M5M 1F7. (See Tables 4.2 and 4.3.) 22

Definition 4.3. Let P be the set of all partial states of robotic arms in a tunnel of width 2. We say that a step of a partial state arrives at the top or bottom border of the tunnel if it touches that border but its preceding step does not touch it.

(a) A partial state of the robot is called irreducible if

• its first step is a horizontal segment along the bottom border of the tunnel, and

• its final step is the first vertical segment or square arriving to the top or bottom border.

(b) A partial state of the robot is called irreducible final if it is empty or

• its first step is a horizontal segment along the bottom border of the tunnel, and

• either

- it never arrives at the top or bottom border, or

- its only arrival to the top or bottom border is at its last step, which is a claw.

Let M and F be the sets of irreducible and irreducible final partial states, respec­ tively.

Let S = n) be the disjoint union of the configuration spaces of all robotic arms of all lengths in width 2. Let B* be the collection of all words that can be made with alphabet B. For instance, a* = {0, a, aa, aaa, aaaa,... } and

{a, b}* = {0, a, b, aa, ab, ba, bb, aaa, aab,... }.

Proposition 4.4. The partial states in S starting with a right step r are in weight- preserving bijection with the words in M*F; that is, each partial state in S corre­ sponds to a unique word of the form mim,2 ■ ■ ■ m ef with nrii G M and f G F. 23

Proof. For each partial state p, let p+ = p and let p~ be the reflection of p across

the horizontal axis. It is clear from the definitions that every partial state that

starts with a horizontal step r factors uniquely as a concatenation m fm * ... m f f ±

where each ra* G M , f € F, and where p± G {p+,p~} for p = m 1,... ,mg,f. It

remains to observe that whether m f is m, or m~, and whether p± is either p or p~,

is determined completely by the previous terms of the sequence. □

Corollary 4.5. If the generating functions for partial states, irreducible partial

states, and irreducible final partial states are C(x,y),M(x,y),F(x,y) respectively, then

1 + xC(x,y) = F(X'V) 1 - M(x, y) ’

Proof. This follows from Proposition 4.4. The extra factor of x comes from the fact

that Proposition 4.4 is counting partial states with an initial right step. □

4.1.3 Enumeration of irreducible partial states

Let us compute the generating function M(x,y) for irreducible partial states.

Proposition 4.6. The generating function for the irreducible partial states M is

x 3 + x4 + 2 x 4y + 2x 5y + x 5y2 + x6y2 M(x,y) ( i - xy

Proof. The word of an irreducible partial state has exactly two symbols that con- 24

Type Illustration Generating function x^y2 Mi = (rr*)D(r*)D ■ J; i (1—x)2 fa * * x6y2 II . M ' (1—x)2 ...... — ... 4 x y M3 = (rr*)D(r*)v . J H ’ (1—x)2 x 6y Mi = (rr*)D(rr*)v' . M (1—x)2 4 x y M5 = (rr*)v(r*)n (1—x)2 K x°y Mg = (rr*)v(rr*)0' 1 * ■ (1—x)2 X 3 M 7 = (rr*)v(r*)v (1—x)2 X 4 M8 = (rr*)v{rr*)v' to 1 — 1------► Table 4.2: The eight types of irreducible partial states and their generating func­ tions.

tribute a vertical move, which can be either a v or a □. Thus there are eight different families Mi,..., Mg, corresponding to the irreducible partial states of the following forms:

... v ... □ ... v ... □' ... v ... v ... v ... v' where □ ' and vl represent a move whose vertical step is in the opposite direction to the previous vertical step. Table 4.2 illustrates these 8 families together with their 25

corresponding generating functions.

Consider for example the family M2 consisting of partial states of the form

...... We must have at least one horizontal step before the first □ , and at

least one horizontal step between the two Ds, to make sure they do not intersect.

Therefore the partial states in M2 are the words in the language (rr*)D(rr*)D',

whose generating function is

2 / 1 \ 2 x 6y2 x-— x)x y {x -— xjxy = ( i - x r

The other formulas in Table 4.2 follow similarly.

Finally, M(x,y) is obtained by adding the eight generating functions in the

table. □

4.1.4 Enumeration of irreducible final partial states

Now let us compute the generating function F(x, y) for irreducible final partial

states.

Proposition 4.7. The generating function for the final irreducible partial states is

1 — x + x 2 + x 2y + x3y + xAy + x 4y2 + x 5y2 F(x, y) = a - x f

Proof. The word of each irreducible final partial state has at most one symbol among

{v. □ }, and can possibly end with l . Again, we let l ' represent a move whose vertical 26

Irreducible move Illustration Generating function 1 Fi = r* 1—x x 2y * II -1 to J to . -1 1 —x x 6y F3 = (rr*)D(r*) (1—x)2 x^y'2 F4 = (rr*)D(r*)L J. 1 (1—a:)2 "j * * x^y2 II » i J 1 (1—x)2 ...... P ...... F6 = (rr*)v(r*) (1- x )2 x6y F7 = (rr*)v(r*)l (1—x)2 x *y4 * * II "i oo J oo (1—x)2

Table 4.3: The eight types of irreducible final partial states and their generating functions.

step is in the opposite direction to the previous vertical step. Thus there are eight different families F i,... , F8, corresponding to the irreducible partial states of the following forms:

...... L

.. .V ...... V ... L . . . V ... l'

. . . □ ...... □ . . . L ...... L'

Table 4.3 shows the eight different families of possibilities together with their 27

corresponding generating functions. To obtain F(x, y) we add their eight generating functions. □

4.1.5 The /-vector and Euler characteristic of «S(i?2,n)

Theorem 4.8. Let

C(x, y) = °n,dXnyd n,d> 0 1 + xy + x 2 + x2y + x 3 + 3 x 3y + x 3y2 + 2 xAy + 2 x 4y2 + xby2 1 — 2 x + x 2 — x3 — x A — 2x 4y — 2 x5y — x 5y2 — xQy2

The above series starts:

C(x,y) = 1 + x(y + 2) + x2(Zy + 4) + x3 (y2 + 8y + 8) + x4(4y2 + 18y + 15)

+x5( l l y2 + 38y + 28) + x6(y3 + 30y2 + 81y + 53) + ... in agreement with Table 4.1.

Proof. This follows from Corollary 4.5 and Propositions 4.6 and 4.7. □

Corollary 4.9. The generating function counting the number cn of states o/

is

cnxn = —^ X — X = 1 + 2x + 4a;2 + 8a:3 + 15a;4 + 28a;5 + 53a;6 -4----- “ 1 - 2a; + x z — x6 — a;4 n>0

Proof. This is a straightforward consequence of Theorem 4.8, substituting y — 0. □

Theorem 4.2. The Euler characteristic of

Proof. The generating function for the Euler characteristic of

,n )x n = ^ ( J](- l)dcM J a;" = C(a;,-1) n> 0 n> 0 \ d > 0 /

1 — X — X 3 + X5 1 — 2a; + x 2 — a;3 + x4 + x5 — x6

= — — — = l JrX-\-X2 -\-X2’-\-... 1 — X by Theorem 4.8. All the coefficients of this series are equal to 1, as desired. □

4.2 Coral PIP

A concise way to describe a position of the robotic arm is by recording the location in the grid of the vertical steps of the robot. This turns out to be useful in identifying a PIP for the robotic arm. We define two important objects which do exactly this: the coral snake and the coral tableau. In particular, special coral tableaux describe the elements in the PIP of the robotic arm, and thus we call the PIPthe coral PIP. 29

Definition 4.10. A coral snake A of height at most m is an oriented path of unit squares, colored alternating black and red (starting with black), inside the tunnel of width m such that:

i The snake A starts at the bottom left of the tunnel, and takes steps up, down,

and right.

ii Suppose A turns from a vertical segment V\ to a horizontal segment H to a

vertical segment V2 at corners C\ and C2. Then V\ and V2 face the same direction

if and only if C\ and C2 have the same color. (Note: we consider the first column

of the snake a vertical segment going up, even if it consists of a single cell.)

Condition (ii) is illustrated in Figure 4.4.

Figure 4.4: Illustration of condition (ii) in Definition 4.10.

The length l(A) is the number of unit squares of A, the height h(A) is the number of rows it touches, and the width w(A) is the number of columns it touches. We say that /x contains A, in which case we write A ■< /i, if A is an initial sub-snake of // obtained by restricting to the first k cells of fi for some k. We write A -< // if A ^ // and A 7^ //. For technical reasons, sometimes we will also consider the empty snake 0. Figure 4.5: A mathematical coral snake and (a photograph of) a real-life coral snake.

Our notion of containment of coral snakes differs from the notion of containment in the plane. For example, if A is the snake with two boxes corresponding to one step right, and ji is the snake with four boxes given by consecutive steps up-right- down, then A is contained in //, in the plane. However, A is not a sub-snake of // and therefore A ^

Definition 4.11. A coral snake tableau (or simply coral tableau) T consists of a coral snake A and a filling of the squares of A with non-negative integers which are strictly increasing horizontally and weakly increasing vertically, following the direction of the snake. We denote by min(T) and max(T) the minimal and maximal entries in T respectively. We call A = sh(T) the shape of T, and we say that T is of type (m ,n ) if h(A) < m and max(T) + /(A) < n.

Definition 4.12. We call a coral tableau tight if its entries are constant along columns and increase by one along rows.

Notice that a tight tableau T is uniquely determined by the shape sh(T) and min­ imal entry min(T). Therefore we denote a tight tableau with sh(T) = A, min(T) = s by the pair (A, s). The empty coral tableau is not tight. 31

4 6 2 3 CM 3 8 11 12 3 5 6 7

1 8 10 13 2 7 8

Figure 4.6: Two coral tableaux; the one on the right is tight.

Lemma 4.13. The possible states of the robotic arm Rm,n are in bijection with the coral tableaux of type (m,n).

Proof. We encode a position P of the robotic arm as a coral tableau T, building it up from left to right as follows. For each new vertical link that the robot has on row i of the grid along the vertical grid line x = j , we add a new square to the row i of the coral tableau and we fill it with the number j. We do this so that the directions of the snake as an oriented path match the directions of the vertical links of the arm. An example is shown in Figure 4.7. Note that the fully horizontal position of the robotic arm corresponds to the empty coral tableau.

It is clear that the entries of T increase weakly in the vertical direction, and increase strictly in the horizontal direction. It remains to check that the snake sh(T) is indeed a coral snake. Suppose sh(T) goes from a vertical V\ to a horizontal

H to a vertical V2, turning at corners C\ and C2 of row i; see Figure 4.4. Assume for definiteness that V\ points up and C\ is red. Then the red and black entries on

H represent up and down steps that the robot takes on row i, respectively; so the direction of V2 is determined by the color of C2 as desired. □ 32

4 6

3 8 11 12

1 8 10 13

Figure 4.7: From a state R to a coral tableau T.

2

► 2 3 4 5 6 7

2 7 8

Figure 4.8: A tight state and its corresponding tight tableau.

Notice that new vertical segments are introduced and existing vertical segments leave the robotic arm at the right. Thus, if' admissible, applying r H>j or lh-P, or flipping the end from vertical to horizontal, brings us closer to the completely horizontal arm. We call these moves descending moves. The states with only a single available descending move are called tight states, and correspond precisely to tight tableaux.

Lemma 4.14. A position P of the robotic arm has only one available descending move to the horizontal position if and only if the corresponding tableau is tight.

Proof Suppose the tableau T is not tight. The last box of any tableau corresponds to a vertical segment in the arm which can be moved right or made horizontal.

Since T is not tight, there exists two consecutive boxes i and * + 1 in T where the difference of the entries xt and xi+i of the boxes is too large. If the two boxes are in the same column, then xJ+1 — x,t > 1 . This implies that after vertical segment i, 33

there is at least one horizontal segment. Thus another move is available. There is no worry of self-intersecting when applying a local move here because the following vertical segment is in a different row. If the box i and * + 1 are in the same row, then x i+i — X{ > 2. This implies that after the ith vertical segment there are at least two horizontal segments before the (i + l)th vertical segment in the same row.

Therefore a second move is available in this scenario as well.

Conversely, if T is tight then the consecutive vertical segments in the same row of P are too close to apply local moves. Similarly the consecutive vertical segments in differing rows have no horizontal segments between them. Therefore the only available move towards the horizontal position is by moving the last vertical segment corresponding the last box of T. □

Definition 4.15. Define the coral PIP Cm>n as follows:

• Elements: tight tableaux (A, s) of a non-empty coral snake A with h(A) < m and a non-negative integer s with s < n — l(A) — w(A) + 1.

• Order: (A, s) < (//, t) if A ■< // and s > t.

• Inconsistency: (A, s) w (/x, t) if neither A nor // contains the other.

In terms of the robotic arm, these tight tableaux are ordered such that a larger element corresponds to a position of the robotic arm that is further awayfrom the completely horizontal arm. Two tight tableaux are comparableonly if for the corresponding positions of the robotic arm, one position is “in between” the other and the horizontal position, this is the shape containment condition. The reverse 34

integer order is describing the fact that a position of the robotic arm that is further away from horizontal should have vertical links closer to the left of the tunnel.

Intuitively, the coral PIP is made up of overlapping sheets, where each sheet is given by a maximal coral snake shape A and all tight tableaux T with sh(T) r< A.

Any pair of elements in a single sheet is consistent, while pairs from two separate sheets are inconsistent. We make use of this decomposition immediately in the proof of the following theorem. Figure 4.9 illustrates the coral PIPs C2,n f°r the tunnel of width 2. Some of the labels are omitted; to obtain them, note that each vertical column consists of the elements (A, s) for a fixed shape A and 0 < s < n — /(A) — w(A) + 1, listed in decreasing order with respect to s.

Theorem 1.1. The configuration space S(Bm n) of the robotic arm Rm,n of length n in a tunnel of width m is a CAT(O) cubical complex. When it is rooted at the horizontal position of the arm, its corresponding PIP is the coral PIP Cni.n of Defi­ nition 4-15.

Proof. We need to show that, when rooted at the horizontal position, S(Rm^n) is the cubical complex X(C'm,n) associated to the PIP Cm

3.4. We proceed in three steps.

Step 1. Decomposing

For a coral snake A, let S{Rm,n)x be the induced subcomplex1 of S(Rm,n) whose vertices are the states of the robot such that their corresponding coral tableaux T

1 Given a polyhedral complex F on vertex set V and a subset U C V, the induced subcomplex of F on U consists of all faces whose vertices belong to U. 35

Figure 4.9: The PIP Cm>n for m = 2 and n — 9. satisfies sh(T) X A. Similarly, let C ^ n be the subPIP of Cm>n consisting of the pairs

(/i, s) such that n -< A. Notice that C xnn is a poset which has no inconsistent pairs.

We have

S(Rm,n)= U S(Rm,n)X Cm,n = U C^n (4.1) A of type (m,n) A of type (m,n) where each (non-disjoint) union is over the coral tableaux A of type (m, n). (Of course, since A -< implies S(Rmtn)x C S(RmtTl)x' and C ^ n C C ^ n, it is sufficient to take the union over those A which are maximal under inclusion.) 36

Step 2. Showing S(Rmtn)x = X (C ^n) for each shape A.

We begin by establishing a bijection between the sets of vertices of both com­ plexes. The vertices of X (C ^ n) correspond to the consistent order ideals of the coral PIP C xt n. Since C ^ n has no inconsistent pairs, these are all the order ideals of C^ n. These order ideals form a poset under inclusion, which is denoted J(C^ n).

The operations of union and intersection make this poset into a lattice, and in fact this lattice is distributive; that is, the join and meet operation satisfy the distributive laws.

On the other hand, the vertices of 5(i?m>n)A correspond to the coral tableaux

T of type (m, n) with shape sh(T) -< A by Lemma 4.13. Let us extend each such tableau T to a tableau of shape A by adding entries equal to oo on each cell of

A — sh(T). We may then identify the set of vertices V(S(Rmin)x) with the resulting set of extended coral tableaux of shape A whose finite entries x satisfy x + /(A) < n.

This set V(S(Rnhn)x) of extended A-tableaux forms a poset under reverse compo­ nentwise order. Note that if Ti and T2 are in V (S(R m^n)x) then the componentwise maximum T\ AT2 and the componentwise minimum T\ VT2 are also in V (

Then the meet A and join V make V(S(Rm.n)x) into a lattice. In fact, the defini­ tions of A and V imply that this lattice is distributive. Birkhoff’s Fundamental

Theorem of Distributive Lattices then states that V(S(Rmtn)x) = J(P) where P is the subposet of join-irreducible elements of V (S (R rn/n)x). Recall the definition of a join-irreducible element in a finite lattice is an element which covers only a single 37

element, by Lemma 4.14 these are precisely the tight tableaux of type (m ,n) and shape A. Thus P = C ^ n.

We conclude that the vertex sets of S (R mtn)x and X (C ^ n) are both in bijection with J(C'^in); that is, the coral tableaux T of type (m,ri) with shape sh(T) < A are indeed in bijection with the consistent order ideals of the coral PIP C ^ n.

We can make the bijection more explicit. Given a coral tableau T of shape n ■< A, we can write T = Ti V T% V • • • V T}^) as follows. For each i let //, be the subsnake consisting of the first i boxes of /x, and let Tt be the unique tight tableau of shape

A, whose ith entry is equal to the zth entry of T. In fact we can reduce this to a minimal equality

T = \ / Ti i jump in T where we say that T jumps at cell i if T remains a coral tableau of type (m, n) when we increase its ith entry by 1. The set A(T) = {Ti : i jump in T} is an antichain in C ^ n, and the bijection above maps the coral tableau T to the order ideal I(T) C C ^ n whose set of maximal elements is / (T )rnax = A(T).

5 6 5 6 v □ 5 6 7 5 6 7 8 Figure 4.10: A coral tableau T = T\ V • • • V T§ = T\ V T2 V T4 V T& as a join of irreducibles.

Having established the bijection between the vertices of S(Rmtn)x and X (C ^ n). we now prove the isomorphism of these cubical complexes. Each cube C(I,M) 38

of X (C ^ n) is given by an order ideal I C C^n n and a subset M C Imax. Let T be the coral tableau corresponding to /, and P the corresponding position of the robotic arm. Then Imax corresponds to the set of descending moves that can be performed at position P to bring it closer to the horizontal position; more precisely, the moves of the form lh-P, or those flipping the end from a vertical position to a horizontal one facing right. An explicit example of such descending moves is illustrated in Figure 4.11.

Any subset M of those moves can be performed simultaneously, so this subset corresponds to a cube in Every cube of

Figure 4.11: The cube corresponding to the moves above.

Step 3. Showing S(Rm,n) = X (C m

Recall that, as A ranges over the shapes of type (m,n), the subcomplexes

S(R m,n)X cover the configuration space S (R m^n), and the subPIPs C^ n (which hap­ pen to have no inconsistent pairs) cover the PIP Cm,n by (4.1). We now claim that the analogous statement holds for the CAT(O) cube complex X (C m>n) as well:

X (C m,n) = (J X (C ^ J . (4.2) X maxi, of type (m,n) 39

To see this, recall that each vertex v of X(C mn. Since I is consistent, either A* ^ A j or A j ■< A i for each pair i,j. Therefore, the maximum shape A among Ai,..., A* contains them all, and is of type (m, n). It follows that v is a vertex of X (C xnn).

Similarly, for any cube C(I, M) of X (C m,n), the consistent order ideal I corresponds to a vertex v in some X (C ^ n), and this means that C(J, M) is in X (C ^ >n) as well.

Since S (R miTl)x — X(C^n) for all A, the last necessary step is to check that the decomposition Sm^n = U S (R m, n)x is compatible with the decomposition X (C rn,n) =

(JA X(C^nn). This follows from the fact that for any A and /i we have

S {R m,nY n = S (R m,„ r , X (C i,n) n X(C',„) = X (C ^ n):

where u = A A /i is the largest coral snake which is less than both A and /i. □

4.3 Enumeration of Tight States in

In Section 4.1 we enumerated the faces of the cubical complex

Corollary 4.9. Having identified the PIP of S(Rm,n) we can now describe any state of the robotic arm in terms of tight tableaux, or equivalently in terms of tight states which form significantly smaller set. Recall that a tight tableau, introduced in Definition 4.12, is a coral tableau whose entries are constant along columns and 40

increase by one along rows. A tight state is simply a state of the robotic arm which corresponds to a tight tableau by the bijection of Lemma 4.13.

Consider the configuration space

/-vector by factorization. Let the weight of a tight state be xn.

Tight states are much simpler than partial states; they produce a word in the alphabet {r,v} where:

• r represents a horizontal segment of the robot facing to the right. Its weight is x.

• v represents a vertical segment. Its weight is x.

1 1

Figure 4.12: The 10 tight states of i?2,4-

By Lemma 4.14, tight states only have a single descending move available, at the last vertical step. Thus, all other vertical steps must be on the same vertical gridlines, or too close to each other for a move to not self-intersect. Figure 4.8 and

Figure 4.12 provide examples of this.

Let A and C be the following two sets of states, A = {rvv, rvrv'} and C =

{0, rv}, where v' is a vertical link in the opposite direction. 41

Proposition 4.16. The tight states in S starting with a right step r then vertical step v are in weight-preserving bijection with the words in A*Cr*; that is each such tight state in S corresponds to a unique word of the form a ^ . ■ ■ aicrr... r with at 6 A and c € C.

Proof. For any state s in i?2,n with an initial horizontal step, let s+ = s, let s~ be the reflection of s across the horizontal axis, and £ {.s+, s~ }.

Let t be a tight state. Because of the conditions on the vertical steps of t, it factors uniquely as a concatenation a f , a f , ...a f, c± , r,... , r where each a,L € A and c € C. The parity of each term is determined by the previous entry of the sequence. □ 3 B EE

Figure 4.13: The factorization of a tight state of i?2,i7- Excluding the first three steps, this is the factorization of Proposition 4.16.

Proposition 4.17. The generating function T(x) counting the number tn of tight states ofS(R.2,n) is

T(x) = t,nx n n > 0 X + X2 + X3

(1 — x)2{l — X3 — X4) = x + 3x2 + 6x 3 + 10x4 + 16x5 + 24x6 + 34x7 + ... 42

Proof. Let t be a tight state. Since a tight state can have arbitrarily many horizontal steps at the beginning, Proposition 4.16 implies t decomposes as r ... ra\a2 ■ ■ ■ aicr... r where a, € A and c € C. Moreover, not all a* and c can be empty since the com­ pletely horizontal state is not tight. Therefore since the states rvv, rvrv'. rv have weights xs,x4,x2, respectively, the generating function of tight states is

. n _ 1 x2 + x3 + x4 __1_ X 2 - ^ n X 1 _ x l _ x3 _ x4 I — X ri> 0

The extra factor of x comes from counting tight states with an initial right step in

Proposition 4.16. □

Corollary 4.18. The number of tight states of Rm^n grows at least exponentially in relation to n for any m > 2.

Proof. Any tight tableaux of type (m, n) is also of type (W , nr) for rn < m', and n < n'. Therefore there is an injective map from the collection of tight states of

Rm,n to a subset of the tight states of Rm/,n'- Since the number of tight states of i?2,n grow exponentially because they are enumerated by a rational generating function, the result follows.

In the case of m = 1, Ardila, Baker, and Yatchak showed that the number of tight states is quadratic [2]. □ 43

Though tight states in i?2,n grow at least exponentially, the rate is significantly less than the growth rate of all states in /?2,n- More formally, an asymptotic formula for the growth rate of a generating function G(x) is given by c- (^)n, where a is the smallest positive root of the denominator of G(x) and c is some constant. In the case of R.2,m Corollary 4.9 tells us that the number of states almost doubles each time we increment the length of the robotic arm since ^ ~ 1.9. Where as Corollary 4.18 implies that the tight states of i?2,n grow by a factor of approximately 1.2. Table

4.4 demonstrates how drastic the difference in quantities is between states and tight states of i?2,n-

Length 5 10 15 20 40 n

States 28 690 16962 416893 ~ 1.3 x 1011 ~ c- 1.9n

Tight 16 85 287 899 ~ 18044 ~ d - 1.2n

Table 4.4: The number of states and tight states, with asymptotics. Both c and d are constants.

This comparison should emphasize the value in determining the coral PIP. As we increase the length of the robotic arm, the number of states in the configuration space quickly becomes too large to reasonably make computations. Instead of working with the configuration space, computations can be made using the PIP, which is much smaller. 44

Chapter 5

Tableaux and Knuth Equivalence

Let 7r, a be two permutations of the symmetric group Sn expressed in one-line no­ tation, and let x < y < z. The two permutations n and a are said to differ by a 1 Knuth relation of the first kind, denoted 7r = cr, if the only difference between the two is in the same three consecutive entries as follows,

7r = 7Ti... y x z ... 7rn and a = 7Ti... yzx ... Trn.

2 Similarly, n and a differ by a Knuth relation of the second kind, denoted 7r = cr, if only in the same three consecutive entries, the two differ as follows

7T = 7Ti... xzy ... 7rn and a = ... zxy ... nn.

K The permutations 7r, a are said to be Knuth equivalent, 7r = cr, if there is a sequence of Knuth related permutations from one to the other. 45

Let n G Sn. Our second reconfigurable system Tn models Knuth equivalence.

K The states of Tn consist of all permutations a € Sn, such that a = tt. There is a move between two permutations if they are Knuth related.

Definition 5.1. The configuration space S(Tn) of the Knuth equivalence class of t t is the following cubical complex. The vertices correspond to permutations which are

Knuth equivalent to tt. There is an edge between vertices o\ and a2 if the two differ by a Knuth relation. The /c-cubes correspond to /c-tuples of commutative Knuth relations which can be applied to a state a. By performing a subset of these k moves we obtain the vertices of the /c-cube.

Figure 5.1: The configuration space

5.1 The Insertion and Recording Tableaux

Knuth relations interact nicely with a family of objects called Young tableaux, but first we must define a Young diagram. A Young diagram is a collection of boxes that are left-justified and whose row lengths are weakly decreasing. The notation for a

Young diagram A is given by the length of each row, A = (Ai, A2,..., A&) where At is the length of row i. If the number of boxes in A is n, this gives us a correspondence between A and a partition of n; we write A I- n.

1 CO 4 2 7 5 8

6

Figure 5.2: A Young diagram of shape A = (3, 2,2,1) and a standard Young tableau of shape A.

We can fill the boxes of a Young diagram with entries from [m] = {1,2,..., m }.

Let A b n. A standard Young tableau T, or simply tableau, is a Young diagram A, which is filled in using every entry in [n] such that the entries increase left to right in each row, and top to bottom in each column.

Suppose a permutation ir € Sn is written in one-line notation. That is tt =

7ri7T2 ... 7rn. Using the entries of 7r, the Robinson-Schensted algorithm [13, 15] con­ structs a sequence of tableaux pairs

(Po, Qo) = (0, 0), (Pi, Ql) • • • , (Pn, Qn) = (P, Q)• 47

The algorithm reads the permutation left to right, inserting x = 1r* into tableau Pi-1 in the following way:

1. Let R be the first row of Pt-\.

2. While x is less than some element of row R. do the following

• Let y be the smallest element of R greater than x. Replace y by x in R.

• Set x := y and let R be the next row down.

3. Now x is greater than every element of R, so place x at the end of R and stop.

TT CO E 7_ _5_ 5 6 1 6 1 2 1 2 1 2 3 4 T _7_ 7 5 5 6 5 6 5 6 8 8 7 7 7 7 OO 8 8 8

1 4 1 4 1 4 7 1 4 7 oo Qi m T Y 2 2 6 2 6 2 6 T T 3 3 3 5 5 5

Table 5.1: The RS algorithm of n = 87561234.

To construct the second sequence of tableaux, let each Qt have the same shape as

Pi. The new box introduced in Qi is filled in with the entry i. The two tableaux

P and Q, or equivalently P(tt) and Q(n), are given the names of insertion tableau 48

and recording tableau, respectively. This algorithm is reversible; when given an ordered pair of tableaux with the same shape A, one can recover the corresponding permutation; for a proof, see [14].

What is special about the RS algorithm for Knuth equivalence is that Knuth equivalent permutations produce the same insertion tableau.

K Theorem 5.2. [11] If n,a G Sn, then n = a if and only if P(n) = P(o).

Figure 5.3: The configuration space 4)), also expressed in terms of tableaux.

Theorem 5.2 implies that each Knuth equivalence class corresponds to a partic­ ular filling of some Young diagram. That is, upon fixing an insertion tableaux P, there is a Knuth equivalent permutation for every possible recording tableau Q of the same shape. These Knuth equivalent permutations are obtained by reversing the RS algorithm.

For the remainder of this chapter we focus on Knuth equivalence classes which correspond to a specific filling of the P tableau. If A b n, we call the canonical 49

equivalence class the class whose P tableau is given by filling in A in order from left to right, and top to bottom. The reconfigurable system T\ is the canonical equivalence class corresponding to shape A.

Let our root be the permutation 7ip obtained by reading the entries of a tableau

P, from left to right, bottom to top. We call up the row word. For example, the row word of the P tableau in Table 5.1 is 87561234. Notice P(ttp) = P, and therefore

77p is contained in the Knuth class defined by P, justifying it as a choice of root.

5.2 The Bump PIP B\

We call the Young diagram A a hook if A b n and A = (n — k, 1,..., 1) = (n — k,lh).

Our main result for this chapter is that the configuration space S(T\) is a CAT(O) cubical complex if and only if A is a hook.

Theorem 5.3. Let T\ correspond to the canonical Knuth equivalence class of shape

A. If X is not a hook, that is if X contains a 2 x 2 square, then S(T\) is not a CAT(O) cubical complex.

Proof. CAT(O) spaces have unique geodesics. With the Euclidean metric, and the fact that the cubes of S(T\) are unit cubes, having a pair of edges connect two vertices implies two geodesics.

Let P be the canonical filling of A, and 7rp be the row word of such a filling.

Suppose the last entry of the second row is k. Since there is a second row with a sec- 50

ond box, somewhere in the permutation np is the sequence, k — 1, k, 1,2. Therefore both kinds of Knuth relations swap k with 1, implying two edges exist between two states in S(T\). Hence there is not a unique geodesic between these two states □

To show the converse of Theorem 1.3, we first define the bump PIP B\.

Suppose A = (n — k, l fc), canonically filled, and let 7r be the corresponding row word. That is, n = n, n — 1,... ,n—k+1,1,2,... ,n—k. Notice that applying a series of Knuth relations to 7r, we may freely push, or bump, entries {n,n— 1,..., n—k+1} to the right as long as we preserve their relative order. As consecutive integers will never be swapped by a Knuth relation, it follows that the Knuth equivalence class of

7r consists of the permutations that have {n, n — 1,..., n — k +1} in decreasing order and {1,2,... ,n — k} in increasing order. Thus one can determine a permutation of the equivalence class merely by locating the positions of either of these sets. We choose to see such a permutation as the set of positions of {n, n— 1 ,... ,n — fc + 1}, a k subset of the n — 1 possible positions (since n — k + 1 cannot be moved into the n-th position). This motivates the following two definitions.

Definition 5.4. Consider the /c-subset, K = {xi < x 2 < ■ ■ ■ < Xk} C [n — 1] and r G [n — 2]. If Xi G {r, r + 1} and {r, r + 1} \xi ^ K, then the bump Brtr+i{i) replaces

Xi with {r, r + 1} \ a;*. We say the value is r + 1.

The bump Br^r+i(i) takes the ith entry out oi n ,n — 1,... ,n — k + 1 in7r and bumps it between position r and position r + 1, if admissible. For example, suppose 51

the row word is irp = 87612345. Two permutations .62,3(1) bumps between are

18273645 and 12873645. That is, # 2,3(1) takes the 1st largest entry and swaps it between positions 2 and 3.

Definition 5.5. Define the bump PIP B\ for A = (n — k. \k) as follows:

• Elements: bumps Brj+i(i) with 1 < i < k and i < r < n — k + i — 2.

• Order: Brjr+i(i) < BSjS+i(j) if j < i and r — s < i — j.

• Inconsistency: There are no inconsistent pairs.

The bump PIP B\ is the collection of all possible bumps of a fc-subset of n— 1. To see the order, begin with {1,2,..., k} as our subset; Br>r+i{i) < BSiS+1(j) if bump

Br,r+i(i) must occur before bump Bss+i(j) is possible. More explicitly, Br>r+i(i) does not have to occur before BStS+i(i) if the difference (r + 1) — (5 + 1) is larger than the number of entries between i and j. Thus r — s < i — j implies the desired bump ordering. Notice that B\ is a product of the two chains (k) x (n — k — 1).

Figure 5.4: The cubical complex of the canonical equivalence class of shape A = (5, l 3), with the corresponding bump PIP. 52

Theorem 1.3. Let T\ be the canonical Knuth equivalence class associated to A. The configuration space S(T\) is a CAT(O) cubical complex if and only if A = (n —k, l fc).

Proof. To complete the proof of Theorem 1.3, consider the canonical equivalence class of A and row word 7r = n, n — 1,..., n — k + 1,1,2,..., n — k as our root.

We show that the PIP corresponding to S(T\) is B\ by referring to the vertices

V(S(T\)) as k-subsets of n — 1. Such a fc-subset {x i < x2 < ■ • • < Xk} denotes the

K positions of the k largest entries of a permutation cr, where a = n.

Let x = {x\ < X2 < • ■ • < Xk} and y = {y\ < y2 < ■ ■ ■ < yk}- Since Knuth relations applied to 7r increment an entry, we can make V (S(T\ j) into a poset by claiming x < y if Xi < y* for all i. In fact, upon defining the two operations, compo­ nentwise maximum and componentwise minimum, V(S(T\)) becomes a distributive lattice. Birkhoff’s representation theorem states that V(S(T\j) is isomorphic to a poset of order ideals J(P ), ordered by inclusion, where P is the set of join-irreducible elements of V(S(T\)).

Since the Knuth relations simply bump entries of the /c-subset, the join-irreducible elements of V(S(T\)) are those subsets with only a single leftward bump available.

Thus V(S(T\)) = J(B\). More explicitly the bump Brr+i(i) corresponds to the join-irreducible element of the form {1,2,..., i — 1, r +1, r + 2,..., r + k — z}, where 53

the value r + 1 is at position i. That is, the entry x3 is as follows

j for j < i

r + j — i + 1 for i < j

We say the bump 6 r,r+i(i) has value r+ 1 at position i. We can recover a fc-subset from an order ideal of bumps by letting entry x3 of the subset be the maximum value of all bumps at position j. If there is no bump with position j, then

1 if j = 1

Xj-1 + 1 if j > 1

For example, if A = (5, l 3), then the maximal antichain (.62,3(1), -85,6(3)} gives the order ideal whose elements are listed in Table 5.2 (see Figure 5.4 for the bump

PIP 6 (5,13)). In this order ideal the maximum value of positions 1,2,3 are 3,4,6, respectively. Thus the corresponding state is (3,4,6}.

Consider an order ideal I of B\ which gives rise to a /c-subset {x\ < x 2 < ■ ■ ■ <

Xk}- The maximal antichain Imax corresponds to the collection of entries which can be decremented by 1. Choosing any d subset M C Imax produces 2d states obtained by removing M from I. We can switch freely between these states, in other words, we can toggle the entries associated to the positions of the bumps. Thus this cube of

X(B\) corresponds to a cube of S(T\). Every cube of S(T\)arises in this way. □ 54

Bump Permutation Subset #3,4(3) 87162345 124 -64,5(3) 87126345 125 -65,6(3) 87123645 126 -62,3(2) 81762345 134 -63,4(2) 81276345 145 -61,2(1) 18762345 234 -62,3(1) 12876345 345

Table 5.2: The elements in the order ideal of (.62,3(1), 65^(3)}.

The set of canonical Knuth equivalence classes is a small family of the collection of all Knuth equivalences classes, but there is a canonical class for every shape A.

If two equivalence classes produce tableaux of the same shape, the permutations in either reconfigurable system are parameterized by the same collection of Q tableaux.

Moreover, a Knuth relation applied to a permutation in positions i, i+l, i+ 2 amounts to a swap of i + 1 with one of the consecutive entries in the Q tableaux [10]. This motivates the following question.

Question 5.6. Let n,a E Sn such that P(n) and P(cr) have thesame shape. Are the configuration spaces S(Tn) and S(Ta) isomorphic?

If the two cubical complexes are isomorphic, then the arguments in Section 5.2 would extend from canonical Knuth equivalences to all others. 55

Chapter 6

Dyck Paths

A Dyck path p is a sequence of 2n line segments in the plane. This path begins at the origin and ends at the point (2n, 0), never falling below the horizontal axis.

Each line segment is one of two types: an upstep (1,1), or a downstep (1, —1). Let up(i) and dp(i) be the number of upsteps and downsteps in the first i terms of p, respectively. The height at i is hp(i) = up[i) — dp(i). Notice that because Dyck paths never fall below the horizontal axis, up(i) > dp(i) for all i. We call the pair, a downstep followed immediately by an upstep, a valley. An upstep immediately followed a downstep is referred to as a peak.

Figure 6.1: A Dyck path of D^ and the local move. 56

The reconfigurable system Dn is the collection of Dyck paths of length 2n with the local move that changes peaks to valleys and vice versa. The cubical complex

S(D n) is the configuration space of Dn.

Definition 6.1. The configuration space S(Dn) of Dyck paths of length 2n is the

following cubical complex. The vertices correspond to the Dyck paths of length 2n.

There is an edge between vertices u and v if the two differ by changing a single peak

to a valley, or vice-versa. The fc-cubes correspond to /c-tuples of disjoint valleys or

peaks, which can be toggled between freely.

6.1 Face Enumeration and the Euler Characteristic of S(Dn)

As we did for the robotic arm, before showing that the configuration space S(Dn)

is a CAT(O) cubical complex, we prove that the Euler characteristic is equal to 1.

Theorem 6.2. The Euler characteristic of the configuration space S(Dn) equals 1.

Recall that the f-vector of a d-dimensional polyhedral complex X is f x =

(/o, f \..... fd) where /& is the number of /c-dimensional faces of X. Moreover the

Euler characteristic x(A) = /o — fi + • • • (—l)d/<2- To prove x (£(£>„)) = 1 we mimic

both the construction of partial states for the robotic arm and a well-known

proof of the enumeration of Dyck paths.

Consider a d-cube in the configuration space of S(Dn). By superimposing the

2d vertices of the cube we get a partial state of the Dyck path. This partial state 57

consists of upsteps, downsteps, and d squares. If any of the d squares of a partial state touch, they can only do so side-by-side. Let the weight of such a partial state be xnyd. The partial states of weight xnyd are in bijection with the (/-cubes of S(Dn).

For an example, see Figure 6.2.

Figure 6.2: A partial state corresponding to a 4-cube in S(Du).

Each partial state gives rise to a unique word in the alphabet {«, d, o } where:

•u represents an upstep. Its weight is x.

•d represents an downstep. Its weight is 1.

•o represents a square, which comes from changing between a peak and a valley. Its weight is xy.

The weight of a partial state is the product of the weights of its symbols. The partial state in Figure 6.2 produces the word w = uuo uud o dddu o duuu o ddd.

After omitting multiplication by 1, the weight of the partial state in this case is

xx(xy)xx(xy)x(xy)xxx(xy) = xuy4.

Definition 6.3. A partial state of a Dyck path is:

• irreducible if it touches the horizontal axis only at the endpoints. 58

• square-irreducible if it only meets the horizontal axis along squares (excluding the

endpoints).

Let I and S be the sets of irreducible and square irreducible partial states,

respectively.

There are two remarks which should be made. First, there is no partial state

with squares as the endpoints, and so a square-irreducible partial state must begin

with an upstep and end with a downstep. Second, the empty partial state is not an

irreducible, or square-irreducible, partial state.

Let T> — U^L0S(Dn) be the disjoint union of the configuration spaces of all Dyck

paths of all lengths. Recall B* is the collection of all words that can be made with

alphabet B.

Proposition 6.4. The partial states in T> are in weight-preserving bijection with

the words in S*.

Proof. A partial state can be decomposed uniquely into a sequence of square- irreducible partial states by breaking apart each downstep and upstep which meet

at, the floor. □

Figure 6.3: The decomposition of Figure 6.2 into square-irreducible partial states. 59

Corollary 6.5. If the generating functions for partial states and square-irreducible

partial states of Dyck paths are P(x,y) and S(x,y) respectively, then

P(x, y) = 1 - S(x,y)'

Similarly we can construct any square-irreducible partial state by joining irre­

ducible partial states together at the ends, adding a square between them. This

results in a second relationship among the partial states.

Proposition 6.6. If I(x,y) is the generating function for irreducible partial states then,

S{x,ycl ) ^ = 1 - yl(x,y)'

Proof. This is a result of the previous statement. The numerator is I(x, y) because

there must at least be one irreducible partial state since the empty partial state

is not square-irreducible. The term yl (x, y) in the denominator is because for any

irreducible partial state that we append, the irreducible state is being joined with a

square. The weight of x this square would contribute is already contributed by the

last downstep and first upstep between the irreducibles that are being joined. □

Proposition 6.7. If P(x,y) andl(x,y) are the generating functions for partial and

irreducible partial states, respectively, then

I(x, y) = xP(x,y). 60

Figure 6.4: How three irreducible states concatenate as a square-irreducible.

Proof. This is immediate because irreducible partial states are partial states which do not reach the floor besides at the ends. Thus given any irreducible partial state can be obtained by adding a segment to the front and end of some partial state.

This contributes a weight of x. □

We are now ready to compute the generating function P(x. y).

Theorem 6.8. Let S(Dn) be the configuration space for Dyck paths of length 2n.

If Pn,d denotes the number of d-dimensional cubes in S(Dn), then

n , ^ n d 1 + x y - y/l - 2zy + x V - 4x r^y) = V =------•

The above series starts as:

P(x,y) = 1 + x + x2(y + 2) + x3(y2 + 5y + 5) + x4(y3 + 9y2 + 2 1y + 14)

+x5(y4 + 14y3 + 56y2 + 84y + 42) + ... 61

Proof. Putting Corollary 6.5 together with Proposition 6.6 and 6.7,

I 1 -yl(x,y) 1 -xyP(x,y) 1 - yl(x, y)~ I (x, y) 1 - xyP{x, y) - xP{x, y)

1 - yl{x,y)

After clearing the denominator and subtracting the remaining terms of the right hand side, we get

(-xy - x)P(x, yf + (1 + xy)P(x, y) - 1 = 0. (*)

Applying the quadratic formula,

1 + xy — — 2 xy + x2y2 — 4x P(x,y) 2x + 2 xy

Notice that if the radical were added, then substituting x = 0 results in a constant term in the numerator and a zero in the denominator, guaranteeing a pole. Thus the root resulting in subtracting the radical provides the correct generating function. □

Corollary 6.9. The generating function counting the number pn of states of S(Dn) is

= 1 + x + 2rr2 + 5x3 + 14x4 + 42x5 + 132a;6 + ... 62

Proof. The result is immediate upon substituting y = 0 into the generating function of Theorem 6.8. This is the generating function for the Catalan numbers [7]. □

Theorem 6.2. The Euler characteristic of the configuration space S (D n) equals 1.

Proof. The generating function for the Euler characteristic of S(Dn) is

' £ x(s(Dn))x" = = P(x,~n 1) n>0 n>0 \d>0 /

Substituting y = — 1 into relation (*),

(x — x)P(x, — l)2 + (1 — x)P(x, — 1) — 1 = 0

P ( x - 1) = 1 1 — X = I + x + x2 + xz +

as desired. □

6.2 Valley PIP

With the movement of changing between peaks and valleys, two Dyck paths of Dn stand out from the rest: the Dyck path with no valleys, and the Dyck path with the largest possible number of valleys. These two Dyck paths are at the opposite ends of the cubical complex S(Dn) and are special because any other Dyck path can be 63

obtained from the first by solely pushing peaks into valleys, and from the second

by purely pushing valleys into peaks. It is for this reason we choose the Dyck path

with no valleys as our home state. We refer to this Dyck path as the upright Dyck

path, and the other as the floor Dyck path.

Figure 6.5: The first five configuration spaces of D n. The upright and floor paths are emphasized.

Choose the upright state to be the home state H in S(Dn). We can define a

partial order on the states of Dn by saying two Dyck paths p, q have the relation p < q if there is an edge-path in S(Dn), of minimal length, from H to q which passes

through p. Since moves away from the upright state H are pushing the peaks into

valleys, this partial order can be viewed as shape containment. In other words two

Dyck paths have the relation p < q if the Dyck path q fits entirely underneath the

Dyck path p. More rigorously, if hp(i) and hq(i) are the heights at position i, then p < q if hg(i) < hp(i) for all i. We refer to this poset as V(S(Dn)), and keep in

mind that such an order relation exists among the vertices. 64

Lemma 6.10. The poset V(S(Dn)) is a distributive lattice.

Proof. Recall a lattice is a poset such that any two elements have a unique join

V (least upper bound) and meet A (greatest lower bound). Let p,q be two Dyck paths of Pn. The meet p A q is the Dyck path r whose height consists of h(i) = max{hp(i), hg(i)} for all i. To see that r is again a Dyck path notice that the height of the endpoints of r are 0, and nowhere is the height negative. Also, the difference between the height of two consecutive entries in r must be 1, otherwise it would contradict p and q being Dyck paths.

To show that r = p A q, suppose there is another greatest lower bound s. For s to be less than p and q, the height at every point of s should be greater than the height at every point of p and every point of q. But this also means that s < r.

The argument for the join is almost identical, replacing max with min.

Recall that a lattice L is distributive if for any x,y,z G L, we have that x A (y V z) = (x A y) V (x A z). Because meet and join check the height of the Dyck path at every point, it is enough to verify distributivity on the height of a single arbitrary entry of the sequence of each Dyck path. Suppose that x, y, z are three Dyck paths of the same length and consider hx(i),hy(i),hz(i) for some i. It is sufficient to check the distributive condition for the two possibilities hx(i) > min{hy(i),hz(i)}, or hx{i) < min{hy(i),hz(i)}.

Case 1. Suppose hx(i) > min{/i2/(i), hz(i)}. Then at position i, x A (y V z) = hx(i).

For the right hand side at position i, one of the meets is hx(i), the other is at least 65

hx{i). The join of these two is again hx(i).

Case 2. Suppose hx(i) < mm{hy(i), hz(i)}. Then the left hand side is simply y\I z at position i. Since the height of x at position i is less than both that of y and z, xAy — y and xAz = z on the right hand side. Therefore (xAy) V (xAz) = yVz. □

We now define the PIP in preparation of showing that S(Dn) is a CAT(O) cubical complex.

Definition 6.11. A valley Dyck path is a Dyck path containing only a single valley.

D efinition 6.12. Define the valley PIP JJn as follows:

• Elements: valley Dyck paths p of length 2n.

• Order: p < p' if hp>(i) < hp(i) for all i.

• Inconsistency: There are no inconsistent pairs.

Proposition 6.13. The number of valley Dyck paths of length 2n is Q ).

Proof. There is a valley Dyck path determined by each lattice point inside the upright Dyck path. For any lattice point the associated valley Dyck path has that

lattice point as the bottom of its only valley. There are (” ) such lattice points. □

Theorem 1.4. The configuration space S(Dn) of Dyck paths of length 2n is a

CAT(O) cubical complex. When it is rooted at the Dyck pathwith no valleys, its

corresponding PIP is the valley PIP Un of Definition 6.12. 66

Figure 6.6: A valley Dyck path in D 7 and the valley PIP U7 .

Proof. Let the upright Dyck path be the root of S(Dn). With the defined order, the set of vertices V(S(Dn)) is a distributive lattice, as shown in Lemma 6.10. By

Birkhoff’s Fundamental Theorem of Distributive Lattices, V(S(Dn)) = J(P) where

P is the subposet of join-irreducible elements of V(S(Dn)). But the join-irreducible elements of V(S(Dn)) are exactly the Dyck paths which only cover one element, that is, the Dyck paths with a single valley. Therefore P = Un and the order ideals of Un correspond to the states of Dn.

So far, we have identified the vertices of the configuration space S(Dn) with the vertices of X(Un), the cubical complex of the valley PIP. What is left to show is that this isomorphism preserves the cubical structure of the two spaces. Consider a cube of X{Un). Such a cube C(I,M) is given by the order ideal / of Un, and a subset M of the maximal antichain of I max- Recall that the maximal antichain of an order ideal is the set of largest elements of /, all of which are incomparable. If I corresponds to a Dyck path p £ V(S(Dn)), then the maximal antichain is the set of valleys of p which can be shifted into peaks. The subset M gives rise to 2'M vertices of S(Dn) which differ by the commutative moves. Any cube of S{Dn) is obtained in this way, that is, a Dyck path p and a collection of valleys. □ 67

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