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 <S(i?2,n) 18 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 <S(i?2,n) 39 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 <S(i?2,n) for arms of length 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 <S(/?2,e) of the robotic arm of length 6 in a 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