Symmetric Cycles 9789814968812, 9781003438328

Citation preview

SYMMETRIC CYCLES

SYMMETRIC CYCLES

Andrey O. Matveev

Published by Jenny Stanford Publishing Pte. Ltd. 101 Thomson Road #06-01, United Square Singapore 307591 Email: [email protected] Web: www.jennystanford.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Symmetric Cycles c 2023 Jenny Stanford Publishing Pte. Ltd. Copyright  All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4968-81-2 (Hardcover) ISBN 978-1-003-43832-8 (eBook)

To the memory of my parents Irina M. Matveeva 1934–2014 Oleg S. Matveev 1931–2022

To the memory of Dr. Damir N. Gainanov 1954–2022

Contents

Preface

1 Preliminaries and Notational Conventions 2 A 2D Perspective on Higher Dimensional Discrete Hypercubes and the Power Sets of Finite Sets 2.1 Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions 2.1.1 The Vertex Set of a Symmetric Cycle Is the Set of Topes of a Rank 2 Oriented Matroid 2.1.2 The Vertices of a Symmetric Cycle with Inclusion-Maximal Positive Parts Are Detected via the Local Maxima of the Cardinalities of Parts 2.1.3 The Subset of Vertices of a Symmetric Cycle with Inclusion-Maximal Positive Parts Is a Critical Committee for the Oriented Matroid H:= (E t , {1, −1}t ) 2.1.4 Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle: Linear Algebraic Decompositions with Respect to a (Maximal Positive) Basis of Rt 2.1.5 Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle: A Poset-Theoretic Interpretation

xiii

1

7 8 8

10

10

12

14

viii Contents

2.1.6 Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle: Decompositions by Means of the Tuples With Inclusion-Maximal Positive Parts in the Set of Reoriented Vertices of the Cycle 2.1.7 The Sizes of Vertex Decompositions: Distance Signals 2.1.8 Basic Metric Properties of Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle 2.1.9 The Negative Parts of Vertices, Graph Distances, and the Scalar Products of Vertices 2.1.10 A Basic Statistic on Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle 2.1.11 Equinumerous Decompositions of a Vertex in a Hypercube Graph with Respect to Its Symmetric Cycles 2.1.12 Change of Cycle, Change of Basis 2.1.13 Circular Translations of Vertex Decompositions 2.2 Coherent Vertex Decompositions in Hypercube  2) and H(t, 2) Graphs H(t, 3 Vertex Decompositions in Hypercube Graphs, and Dehn–Sommerville Type Relations 3.1 Dehn–Sommerville Equations for the Feasible Subsystems of Rank 2 Infeasible Systems of Homogeneous Strict Linear Inequalities 3.2 Dehn–Sommerville Type Relations for Vertex Decompositions in Hypercube Graphs. I 4 Vertex Decompositions in Hypercube Graphs, and Orthogonality Relations 4.1 Dehn–Sommerville Type Relations for Vertex Decompositions in Hypercube Graphs. II 4.2 Orthogonality Relations for Vertex Decompositions in Hypercube Graphs

16 18

21 23

28

29 32 34 34

39

40 43

49 50 52

Contents

5 Distinguished Symmetric Cycles in Hypercube Graphs and Computation-free Vertex Decompositions 5.1 Distinguished Symmetric Cycles in Hypercube Graphs 5.2 Separation Sets, Negative Parts, and Decompositions of Vertices of Hypercube Graphs 5.3 The Sizes of Decompositions 5.4 Equinumerous Decompositions of Vertices 5.5 Decompositions, Inclusion–Exclusion and Valuations

6 Distinguished Symmetric Cycles in Hypercube Graphs and Pairwise Decompositions of Vertices: Two-member Families of Disjoint Sets 6.1 Partitions of the Negative Parts of Vertices into Two Subsets, and Decompositions of Vertices 6.2 Inclusion of the Negative Parts of Vertices, and Decompositions of Vertices 6.3 Statistics on Partitions of the Negative Parts of Vertices of Hypercube Graphs and on Decompositions of Vertices

7 Distinguished Symmetric Cycles in Hypercube Graphs and Pairwise Decompositions of Vertices: Arbitrary Two-member Clutters 7.1 Statistics on Unions of the Negative Parts of Vertices of Hypercube Graphs and on Decompositions of Vertices 7.1.1 Vertex Pairs of the Discrete Hypercube {1, −1}t 7.1.2 Case: |A ∩ B| = 0, and |A ∪˙ B| = t 7.1.3 Case: |A ∩ B| = 0, and |A ∪˙ B| < t 7.1.4 Case: |A ∩ B| > 0, and |A ∪ B| = t 7.1.5 Case: | A ∩ B| > 0, and |A ∪ B| < t 8 Vertices, Their Relabeled Opposites, and Distinguished Symmetric Cycles in Hypercube Graphs 8.1 Vertices, Their Relabeled Opposites, and Decompositions

57 59 61 82 84 86

89 90 93

95

117 120 120 122 123 125 148

207 209

ix

x Contents

9 Set Families, Blocking Sets, Blockers, and Distinguished Symmetric Cycles in Hypercube Graphs 9.1 Blocking Sets and Blockers 9.2 Increasing Families of Blocking Sets, and Blockers: Set Covering Problems 9.3 (Sub)set Families: Characteristic Vectors and Characteristic Covectors 9.4 Increasing Families of Blocking Sets, and Blockers: Characteristic Vectors and Characteristic Covectors 9.4.1 A Clutter {{a}} 9.4.1.1 The principal increasing family of blocking sets B({{a}}) = {{a}} 9.4.1.2 The blocker B({{a}}) = {{a}} 9.4.1.3 More on the principal increasing family B({{a}}) = {{a}} 9.4.2 A Clutter { A} 9.4.2.1 The increasing family of blocking sets B({A}) = {{a} : a ∈ A} 9.4.2.2 The blocker B({A}) = {{a} : a ∈ A} 9.4.2.3 More on the increasing families {A} and B({A}) 9.4.3 A Clutter A := {A 1 , . . . , A α } 9.4.3.1 The increasing family of blocking sets B(A) 9.4.3.2 The blocker B(A) 9.4.3.3 More on the increasing families A and B(A) 9.4.3.4 The characteristic vector of the subfamily of inclusion-minimal sets min F in a family F 9.4.3.5 More on the blocker B(A) 9.5 Decompositions of the Characteristic (Co)vectors of Families 9.5.1 A Clutter {{a}} 9.5.2 A Clutter {A} 9.5.3 A Clutter A := {A 1 , . . . , A α }

221 226 229 233 236 238 238 239 239 243 243 245 245 247 247 248 248

250 251 253 261 263 264

Contents

10 Vertex Decompositions and Subtope Decompositions in Hypercube Graphs 10.1 Symmetric Cycles in Hypercube Graphs: Vertices, Edges and Subtopes 10.2 Vertex Decompositions and Subtope Decompositions in Hypercube Graphs with Respect to the Edges of Their Symmetric Cycles 10.3 Vertex Decompositions in Hypercube Graphs with Respect to the Edges of Their Distinguished Symmetric Cycles

277

Appendix A: Enumeration of Smirnov Words over Three-letter and Four-letter Alphabets A.1 Ternary Smirnov Words A.2 Smirnov Words over a Four-letter Alphabet

281 282 284

Appendix B: The Increasing Families of Sets Generated by Self-dual Clutters B.1 Long f - and h-Vectors of Set Families B.2 Abstract Simplicial Complexes , Such That ∗ = , on Their Ground Sets Et of Even Cardinality t B.3 The Increasing Families of Sets Generated by Self-Dual Clutters on Their Ground Sets Et of Even Cardinality t B.4 Remarks on Set Families F Such That F ∗ = F

267 268

271

287 289

292

296 297

Bibliography

301

List of Notation

313

Index

319

xi

Preface

The present monograph begins, in a sense, where the book Pattern Recognition on Oriented Matroids, published with De Gruyter in 2017, left off. Given an ordered two-letter alphabet (θ, α) and an integer dimensional parameter t, the discrete hypercube {θ, α}t is the vertex set of the hypercube graph Γ (t, 2; θ, α), a central discrete mathematical construct with countless applications in theoretical computer science, Boolean function theory, combinatorics, combinatorial optimization, coding theory, discrete and computational geometry, etc. Let E t := [t] := {1, . . . , t} denote a ground set of indices. A pair {X , Y } of vertices X := (X (1), . . . , X (t)) and Y of the hypercube {θ, α}t by definition is an edge of the graph Γ (t, 2; θ, α) if the Hamming distance |{e ∈ E t : X (e) = Y (e)}| between the ordered tuples X and Y is 1. A vertex Z ∈ {θ, α}t of the hypercube graph Γ (t, 2; θ, α) has its ‘positive part’ p(Z ) := Z + , defined to be the subset {e ∈ E t : Z (e) = θ} ⊆ E t , and its ‘negative part’ n(Z ) := Z − := {e ∈ E t : Z (e) = α} = E t − p(Z ). Recall that the power set 2[t] of the set E t is defined to be the family {A : A ⊆ E t } of all subsets of E t . The partial ordering of this family by inclusion turns the power set 2[t] into the Boolean lattice B(t) of rank t. Throughout the monograph, by convention we interpret the power set   2[t] = n(Z ) : Z ∈ {θ, α}t of the ground set E t as the family of the negative parts of vertices of a particular discrete hypercube, {1, −1}t ⊂ Rt , or {0, 1}t ⊂ Rt , associated with the two-letter alphabet (θ, α) := (1, −1),

xiv

Preface

or (θ, α) := (0, 1), respectively, where the letters θ and α are regarded as real numbers. In the monograph, we consider both the hypercube graphs H (t, 2) := Γ (t, 2; 1, −1) ,

 (t, 2) := Γ (t, 2; 0, 1) , and H

though our main interest lies in the graph H (t, 2) on its vertex set {1, −1}t . A symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H (t, 2) is defined to be a 2t-cycle, with its vertex set V(D) := {D0 , D1 , . . . , D2t−1 }, such that Dk+t = −Dk , 0 ≤ k ≤ t − 1 . The vertex set V(D) of the symmetric cycle D is a maximal positive basis of the space Rt . For any vertex T ∈ {1, −1}t of the graph H (t, 2) there exists a unique inclusion-minimal and linearly independent subset (of odd cardinality) Q (T , D) ⊂ V(D) such that  Q. T = Q∈ Q (T , D)

In particular, that linear algebraic decomposition describes how the members of the family of subsets {n(Q) : Q ∈ Q (T , D)} ⊂ 2[t] vote for, or against, the elements of the ground set E t , thus arriving at their collective decision on the subset n(T ) ⊆ E t . Informally speaking, we discuss in the monograph various aspects of how (based on the above decomposition) all the vertices of the discrete hypercube {1, −1}t , as well as the power set of the ground set E t , emerge from a rank 2 oriented matroid, from an underlying rank 2 system of linear inequalities, and thus literally from an arrangement of t distinct straight lines crossing a common point on a piece of paper. In the introductory Chapter 2, we briefly recall basic properties of vertex decompositions in hypercube graphs H (t, 2) with respect to their symmetric cycles. We then establish a connection between co (t, 2). herent decompositions in the hypercube graphs H (t, 2) and H In Chapter 3, we recall some enumerative results on rank 2 infeasible systems of linear inequalities related to arrangements of oriented lines in the plane. Dehn–Sommerville type relations are presented that concern the numbers of faces of abstract

Preface

simplicial complexes associated with large-size decomposition sets for vertices of hypercube graphs. Chapter 4 concerns additional Dehn–Sommerville type relations that are valid for large-size decomposition sets for vertices in hypercube graphs. We present a common orthogonality relation that establishes a connection between enumerative properties of largesize decomposition sets in the graphs H (s, 2) and H (t, 2) with specific dimensional parameters s and t. In Chapter 5, we give a few comments on certain distinguished symmetric cycles in hypercube graphs. The main results of the chapter relate to the interval structure of the negative parts of vertices of hypercube graphs, to computation-free decompositions with respect to the distinguished symmetric cycles, and to statistics on decompositions. We also discuss equinumerous decompositions of vertices. We conclude the chapter by mentioning that vector descriptions of vertex decompositions with respect to arbitrary symmetric cycles are valuations on the Boolean lattices of subsets of the vertex sets of hypercube graphs. In Chapter 6, we touch on the question on a structural connection between the decomposition sets for vertices whose negative parts are comparable by inclusion. Further enumerative results concern statistics on partitions of the negative parts of vertices of hypercube graphs and on decompositions of vertices. The key computational tool that allows us to present quite fine statistics is an approach to enumeration of ternary Smirnov words (i.e., words over a three-letter alphabet, such that adjacent letters in the words never coincide) discussed in Appendix A. An even more involved analysis, based on enumeration of Smirnov words over four-letter alphabets (also discussed in Appendix A), leads us in Chapter 7 to statistics on unions of the negative parts of vertices of hypercube graphs and on decompositions of vertices. An innocent-looking transformation (with serious applications given later in Chapter 9) of vertices of a discrete hypercube {1, −1}t , that turns a vertex of the hypercube graph H (t, 2) into its ‘relabeled opposite’, is presented and discussed in Chapter 8. Recall that a nonempty family of nonempty subsets A := {A 1 , . . . , A α } ⊂ 2[t] of the ground set E t is called a clutter (Sperner

xv

xvi

Preface

family), if no set from the family A contains another. One says that a subset B ⊆ E t is a blocking set of the clutter A if the set B has a nonempty intersection with each member A i of A. The blocker B(A) of the clutter A is defined to be the family of all inclusion-minimal blocking sets of A. In Chapter 9, we drastically change the dimensionality of our research constructs from t to 2t , since we indirectly represent the families of blocking sets of clutters as the negative parts of relevant vertices of the  (2t , 2) associated with the discrete hypercube graphs H (2t , 2) and H 2t 2t hypercubes {1, −1} and {0, 1} , respectively. We describe in detail a ‘blocking/voting’ connection between the families of blocking sets of the clutters A and B(A). In Appendix A, Smirnov words over three-letter and four-letter alphabets are enumerated. In Appendix B we investigate the enumerative properties of the subset families generated by the self-dual clutters A = B(A). Andrey O. Matveev February 2023

Chapter 1

Preliminaries and Notational Conventions

Throughout the monograph, ‘:=’ means equality by definition. – Consider an arrangement of distinct straight lines {x ∈ R2 : ae , x = 0, e ∈ E t }

(1.1)

in the plane R2 (throughout the monograph, v, w means the  standard scalar product e ve we of real vectors v and w of relevant dimension), with the set of corresponding normal vectors A := {ae : e ∈ E t } . One associates with the arrangement (1.1) the rank 2 system of homogeneous strict linear inequalities {ae , x > 0 : x ∈ R2 , ae ∈ A} . The lines of the arrangement are oriented: a vector v := (v1 , v2 ) ∈ R2 − {0} lies on the positive side of a line L e := {x ∈ R2 : ae , x = 0} if we have ae , v > 0. Similarly, a region T of the arrangement (1.1),  that is, a connected component of the complement R2 − e∈ E t L e , lies on the positive side of the line L e if we have ae , v > 0, for an arbitrary vector v ∈ T . Symmetric Cycles Andrey O. Matveev c 2023 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4968-81-2 (Hardcover), 978-1-003-43832-8 (eBook) www.jennystanford.com

2 Preliminaries and Notational Conventions

The arrangement (1.1) realizes a rank 2 oriented matroid N in the following manner (see, e.g., Example 7.1.7 in Ref. [1]): For a vector v ∈ R2 , the ordered tuple of signs X := (sign(ae , v) : e ∈ E t ) ∈ {1, 0, −1}t := {‘+’, ‘0’, ‘−’}t is called a covector of N . The covectors T ∈ T ⊂ {1, −1}t of the oriented matroid N are called its topes (maximal covectors), and the set of topes T is in a one-to-one correspondence with the set of regions of the arrangement (1.1). The cocircuits C ∗ ∈ {1, 0, −1}t of the oriented matroid N := (E t , T ), on the ground set E t , and with its set of topes T , are the covectors that correspond to the rays emanating from the origin; one sign component of each cocircuit is 0. – Given a vector z := (z1 , . . . , zt ) ∈ Rt , we denote its support {e ∈  0} by supp(z). E t : ze = – Now consider the rank t oriented matroid H := (E t , {1, −1}t ) on the ground set E t , and with its set of topes (maximal covectors) T := {1, −1}t , realizable (see, e.g., Example 2.1.4 in Ref. [1]) as the arrangement of coordinate hyperplanes  {x : = (x1 , . . . , xt ) ∈ Rt : | supp(x)|  (1.2) = t − 1, xe = 0} : e ∈ E t in the space Rt . The hyperplanes of the arrangement are oriented: a vector v := (v1 , . . . , vt ) ∈ Rt − {0} lies on the positive side of a hyperplane H e := {x ∈ Rt : | supp(x)| = t − 1, xe = 0} if ve > 0. Similarly, a region T of the arrangement (1.2), that is, a connected component  of the complement Rt − e∈E t H e , lies on the positive side of the hyperplane H e if ve > 0, for an arbitrary vector v ∈ T . For a vector v ∈ Rt , the sign tuple X := (sign(ve ) : e ∈ E t ) ∈ {1, 0, −1}t := {‘+’, ‘0’, ‘−’}t is a covector of the oriented matroid H. The cocircuits C ∗ ∈ {1, 0, −1}t of H are the covectors that have one sign component different from 0. – If M is one of the above oriented matroids N := (E t , T ) and H := (E t , T := {1, −1}t ), then a sign tuple S := (S(1), . . . , S(t)) ∈ {1, 0, −1}t , with exactly one zero component S(i ) = 0, is called a subtope of M if there are two topes, T  := (T  (1), . . . , T  (t)) ∈ T , and T  ∈ T , such that the Hamming distance between the tuples T  and T  is 1, that is, |{e ∈ E t : T  (e) = T  (e)}| = 1, and T  (i ) = T  (i ).

Preliminaries and Notational Conventions

Note that the subtopes of the rank 2 oriented matroid N are its cocircuits. The separation set S(T  , T  ) of topes T  and T  is defined by S(T  , T  ) := {e ∈ E t : T  (e) = T  (e)} , and the graph distance between them is d(T  , T  ) := |S(T  , T  )| , that is, d(T  , T  ) is the Hamming distance between the tuples T  and T  . Recall that 1 1 d(T  , T  ) = t − T  + T  2 = T  − T  2 4 4 1   = (t − T , T ) , 2 where X 2 := X , X . If t is even, then we have1 t . 2 – The vertex set of the tope graph of an oriented matroid M by definition is its set of topes T . Topes T  and T  are adjacent in the tope graph if they have a common subtope. Note that the tope graph of the oriented matroid N is a cycle on its vertex set T . The tope graph of the oriented matroid H is the hypercube graph on its vertex set {1, −1}t . The tope graph turns into the Hasse diagram of the tope poset of the oriented matroid M, based at a tope B ∈ T , if we partially order the set of topes as follows: T  , T   = 0 ⇐⇒ d(T  , T  ) =

T  T  ⇐⇒ S(B, T  ) ⊆ S(B, T  ) . The graph distance d(B, ·) becomes in this situation the rank function of the tope poset of the oriented matroid M. – In the present monograph, we use almost everywhere the nonstandard representation of sign components ‘+’, ‘0’ and ‘−’ of covectors/vertices by the real numbers 1, 0 and −1, respectively. Nevertheless, most figures involve the traditional notation. subsets of vertices of discrete hypercubes {1, −1}t , with zero pairwise scalar products, are the rows of Hadamard matrices, see, e.g., Refs. [2, 3, 4].

1 Interesting

3

4 Preliminaries and Notational Conventions

As a common rule, no matter what the notation is used, covectors/vertices are always thought of as elements of the real t Euclidean space Rt (or R2 , in Chapter 9) of row vectors. – For readability, we prefer the nonstandard notation p(X ) and n(X ) for the positive parts and negative parts of tuples X ∈ {1, −1}t , instead of the traditional notation X + and X − , respectively. Thus, by convention we define the parts by p(X ) : = {e ∈ E t : X (e) =

1} =: X + ,

and n(X ) : = {e ∈ E t : X (e) = −1} =: X − . – We denote by T(+) the row tuple of all one’s T(+) := (+, . . . , +) := (1, . . . , 1) ∈ Rt . If T(+) is a (maximal) covector of our oriented matroid M, then it is called the positive tope of M. We say that T(+) is the positive vertex of the discrete hypercube {1, −1}t , and that it is the positive vertex of the hypercube graph H (t, 2). The row tuple T(−) is the element T(−) := −T(+) := (−, . . . , −) := (−1, . . . , −1) of the space Rt ; if the tuple T(−) is a (maximal) covector of the oriented matroid M, then T(−) is called the negative tope of M. – A subset A ⊆ E t of the ground set of the oriented matroid N is called acyclic if there is a tope F ∈ T of N such that p(F ) ⊇ A. – Given a tuple T ∈ {1, −1}t , and a subset A ⊆ E t , the notation −A T is used to denote the tuple obtained from T by sign reversal or reorientation on the set A:  −T (e) , if e ∈ A, (−A T )(e) := T (e) , if e ∈ A. Thus, by convention we have n(−A T(+) ) := A. One says that the oriented matroid −A N , whose set of topes by definition is the set −A T := {−A T : T ∈ T }, is obtained from the oriented matroid N by reorientation on the subset A. Since we have −A {1, −1}t = {1, −1}t and, as a consequence, −A H = H, the oriented matroid H is insensitive to reorientations.

Preliminaries and Notational Conventions

– The unit vectors of the standard basis of the space Rt are denoted by σ (i ) := (0, . . . , 1, . . . , 0), 1 ≤ i ≤ t. ↑

i

– We denote by #A the number α of sets in a family A := { A 1 , . . . , A α }. The cardinality of a finite set A is denoted by | A|. – An abstract simplicial complex  on its vertex set E t is defined to be a family  ⊆ 2[t] , such that {e} ∈ , for every vertex e ∈ E t , and the following implications hold: A, B ⊆ E t , A ⊆ B ∈ 

=⇒

A∈.

Members of the family  are called faces. The inclusion-maximal faces are the facets of the complex .

5

Chapter 2

A 2D Perspective on Higher Dimensional Discrete Hypercubes and the Power Sets of Finite Sets

The power set 2[t] of the t-element set E t := [t] := {1, . . . , t}, that is, the family of all subsets of the set E t , emerges in a somewhat miraculous manner, in a 2D context,1 from t distinct straight lines crossing a common point on a piece of paper. Indeed, on the one hand, the sets composing the family 2[t] can be seen as the negative parts of vertices T ∈ {1, −1}t of the hypercube graph H (t, 2). On the other hand, the sequence of sign tuples describing the regions of our arrangement of oriented lines in the plane R2 can be  D) of a symmetric cycle D in interpreted as the vertex sequence V(  D) ⊂ {1, −1}t is an ordered the graph H (t, 2). Since the sequence V( t maximal positive basis of the space R , any vertex T ∈ {1, −1}t of the graph H (t, 2) admits its expansion into a unique inclusion-minimal and linearly independent subset (of odd cardinality) of the vertex set V(D).

1 2D means 2-dimensional.

Symmetric Cycles Andrey O. Matveev c 2023 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4968-81-2 (Hardcover), 978-1-003-43832-8 (eBook) www.jennystanford.com

8 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

In Section 2.1 of this introductory chapter we briefly recall basic properties of vertex decompositions in hypercube graphs H (t, 2) with respect to their symmetric cycles. In Section 2.2 we establish a connection between coherent decompositions in the hypercube  (t, 2) and H (t, 2) whose vertex sets are the sets {0, 1}t graphs H and {1, −1}t , respectively.

2.1 Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions Given the hypercube graph H (t, 2) on its vertex set {1, −1}t , a symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the graph H (t, 2), with its vertex sequence2  D) : = (D0 , D1 , . . . , D2t−1 ) , V(

(2.1)

and with its vertex set V(D) : = {D0 , D1 , . . . , D2t−1 } ,

(2.2)

is defined to be a 2t-cycle such that Dk+t = −Dk , 0 ≤ k ≤ t − 1 .

(2.3)

This section gives an overview of basic properties of the vertex decompositions in hypercube graphs H (t, 2) with respect to their symmetric cycles.

2.1.1 The Vertex Set of a Symmetric Cycle Is the Set of Topes of a Rank 2 Oriented Matroid As noted in Remark 1.7 of Ref. [5], the vertex set V(D) of a symmetric cycle D in a hypercube graph H (t, 2) is the set of topes (i.e., maximal covectors) of a rank 2 oriented matroid denoted as H D . The ground set of H D is the ground set E t of the oriented matroid H := (E t , {1, −1}t ). 2 Figure

2.6 on page 19 explains why we should make a distinction between vertex sequences and vertex sets.

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.1 (a) A symmetric cycle D in the hypercube graph H ((t := 3), 2). The graph H (t, 2) is the tope graph of the oriented matroid H := (E t , {1, −1}t ), on the ground set E t , with its set of topes {1, −1}t . (b) A central arrangement of t := 3 oriented lines {L 1 , L 2 , L t } that realizes the rank 2 oriented matroid H D := (E t , V(D)), on the ground set E t , with its set of topes V(D). The positive sides of lines are marked by black triangles. The regions of the arrangement are labeled with the vertices of the cycle D, that is, with the topes of the oriented matroid H D .

The set of cocircuits of the oriented matroid H D is the subset of subtopes of H, each of which is covered in the big face lattice of the oriented matroid H by a pair of adjacent vertices of the cycle D. The rank 2 oriented matroid H D can be represented by a central line arrangement in the plane.3 The set V(D) of topes of H D corresponds to the centrally symmetric set of regions of the arrangement. The set of cocircuits of H D corresponds to the centrally symmetric set of rays emanating from the origin; see Example 7.1.7 in Ref. [1]. Example 2.1. Suppose t := 3, and consider a symmetric cycle D in the hypercube graph H (t, 2), depicted in Figure 2.1(a). The vertex set V(D) of the cycle D is the set of topes of a rank 2 oriented matroid H D realized in the plane as a central arrangement of oriented lines depicted in Figure 2.1(b). 3 By

convention, each line of a central arrangement crosses the origin 0 := (0, 0) of the plane R2 .

9

10 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

2.1.2 The Vertices of a Symmetric Cycle with Inclusion-Maximal Positive Parts Are Detected via the Local Maxima of the Cardinalities of Parts Given a symmetric cycle D in a hypercube graph H (t, 2), we denote by max+ (V(D)) the subset of all vertices T ∈ V(D) that have inclusion-maximal positive parts. Note that the family {p(T ) : T ∈ max+ (V(D))} ⊂ 2[t] is a clutter, that is, its members are pairwise incomparable by inclusion. Let (T k1 , T k2 , T k3 ) be a 2-path in the symmetric cycle D of the graph H (t, 2), where T k1 = T k3 . We have T k2 ∈ max+ (V(D)) ⇐⇒ | p(T k1 )| = | p(T k3 )| = | p(T k2 )| − 1 ; see Remark 1.8(ii) in [5]. In other words, let { f } := S(T k1 , T k2 ), and {g} := S(T k2 , T k3 ). Then the inclusion T k2 ∈ max+ (V(D)) holds if and only if T k2 ( f ) = T k2 (g) = 1. Example 2.2. Suppose t := 3, and consider a symmetric cycle D in the hypercube graph H (t, 2), depicted in Figure 2.2. The subset of vertices of the cycle D that have locally maximal cardinalities of their positive parts is precisely the subset max+ (V(D)) of vertices of the cycle D with inclusion-maximal positive parts. Note that for a vertex T k2 ∈ V(D), the related separation sets are S(T k1 , T k2 ) = {1}, and S(T k2 , T k3 ) = {3}. Since T k2 (1) = T k2 (3) = 1, we have the inclusion T k2 ∈ max+ (V(D)).

2.1.3 The Subset of Vertices of a Symmetric Cycle with Inclusion-Maximal Positive Parts Is a Critical Committee for the Oriented Matroid H:= ( E t , {1, −1}t ) According to Definitions 1.1(i) and 1.3(ii) of Ref. [5], an inclusionminimal subset K∗ ⊂ {1, −1}t , such that  K = T(+) , (2.4) K ∈K∗

is called a critical committee for the oriented matroid H := (E t , {1, −1}t ).

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.2 A symmetric cycle D in the hypercube graph H ((t := 3), 2). The vertices of the cycle are labeled with the cardinalities of their positive parts. Since the cardinalities of the positive parts of vertices T k2 , F and G are locally maximal (e.g., we have | p(T k1 )| = | p(T k3 )| = | p(T k2 )| − 1), this guarantees that these three vertices are the only vertices that belong to the vertex set of the cycle D and have inclusion-maximal positive parts, that is, max+ (V(D)) = {T k2 , F , G}. subset of the set {1, −1}t , with The set max+ (V(D)) is an inclusion-minimal  (+) the property T := (1, 1, 1) = K∈max+ (V(D)) K = (1, −1, 1) + (1, 1, −1) + (−1, 1, 1), that is, max+ (V(D)) by definition is a critical committee for the oriented matroid H := (E t , {1, −1}t ).

Returning to our symmetric cycle D of the hypercube graph H (t, 2), described in (2.1)(2.3), recall that the subset max+ (V(D)) ⊂ V(D) is a critical committee for H; see Proposition 1.27 in Ref. [5]. The set max+ (V(D)) contains an odd number of vertices. For each element e ∈ E t of the ground set of the oriented matroid H we have

| max+ (V(D))| |{K ∈ max+ (V(D)) : K (e) = 1}| = . 2 Example 2.3. Suppose t := 3, and consider again the symmetric cycle D in the hypercube graph H (t, 2), depicted in Figure 2.2. Since the set K∗ := max+ (V(D)) is an inclusion-minimal subset of the set {1, −1}t with the property (2.4), the set max+ (V(D)) by definition is a critical committee for the oriented matroid H := (E t , {1, −1}t ). As noted in Remark 1.28 of Ref. [5], a simple linear algebraic argument given in Section 11.1 of Ref. [5] and recalled below in Subsection 2.1.4 guarantees that the set max+ (V(D)) is the only

11

12 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

critical committee whose topes all belong to the vertex set V(D) of the symmetric cycle D. Transformations of critical tope committees under one-element reorientations of the vertex sets of symmetric cycles are discussed in Section 1.3 of Ref. [5].

2.1.4 Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle: Linear Algebraic Decompositions with Respect to a (Maximal Positive) Basis of Rt For a symmetric cycle D in a hypercube graph H (t, 2), described in (2.1)(2.3), the square matrix ⎛ 0 ⎞ D ⎜ D1 ⎟ ⎜ . ⎟ ⎜ ⎟ (2.5) M := M(D) := ⎜ .. ⎟ ∈ Rt×t ⎜ t−2 ⎟ ⎝D ⎠ Dt−1 is nonsingular, since we have 0 = | det(M)| = 2t−1 ; see Section 11.1 of Ref. [5]. The vertex sequence (D0 , D1 , . . . , Dt−1 ) of the corresponding (t − 1)-path in the graph H (t, 2) is an ordered  D) of basis of the space Rt . This means that the vertex sequence V( the symmetric cycle D is an ordered maximal positive basis of Rt ; cf. Theorem 6.4 in Ref. [6]. Given an arbitrary vertex T ∈ {1, −1}t of the graph H (t, 2), we thus have T = x·M, for the row vector x := x(T ) := x(T , D) = (x1 , . . . , xt ) := T · M−1 .

(2.6)

If T := − A T(+) , for some subset A ⊆ E t , then, instead of x (−A T(+) , D), we often use the notation x( A) := x(A, D) .

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

While the vertex −A T(+) of the graph H (t, 2) can be viewed as the ‘characteristic covector’ of the subset A ⊆ E t , the ‘x-vector’ −A T(+) · M−1 is yet another linear algebraic portrait of the set A. Note that x ∈ {−1, 0, 1}t . For any vertex T ∈ {1, −1}t of the hypercube graph H (t, 2), there exists a unique inclusion-minimal subset Q (T , D) ⊂ V(D), such that  Q. (2.7) T = Q∈ Q (T , D)

Sometimes we let q(T ) := q(T , D) denote the cardinality of the set Q (T , D): q(T ) := |Q (T , D)| . Moreover, if T := −A T , for some subset A ⊆ E t , we often use for the quantity |Q (T , D)| the notation (+)

q(A) := q(A, D) . The subset Q (T , D) = {xi (T ) · Di −1 : xi (T ) = 0} ⊂ Rt ,

(2.8)

of odd cardinality, is linearly independent. We certainly  have Q∈Q (T , D) T , Q = t. In the particular case, where T is the positive vertex T(+) , the set Q (T(+) , D) = {xi (T(+) ) · Di −1 : xi (T(+) ) = 0} = max+ (V(D)) is a critical committee for the oriented matroid H := (E t , {1, −1}t ). Example 2.4. Let us return to the symmetric cycle D in the hypercube graph H ((t := 3), 2), considered earlier in Example 2.3; see Figure 2.3. The t × t matrix M(D) defined by (2.5) is the nonsingular matrix ⎛ ⎞ −1 1 −1 M := M(D) = ⎝ 1 1 −1⎠ . 1 1 1 The sequence of its rows, that is, the vertex sequence of the (t − 1)path (D0 , D1 , Dt−1 ) in the cycle D, is an ordered basis of Rt .

13

14 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

Figure 2.3 A symmetric cycle D in the hypercube graph H ((t := 3), 2). The only inclusion-minimal subset of vertices of the cycle D, such that their sum is the vertex T of the graph H (t, 2), is the set Q (T , D) given in (2.8). Here Q (T , D) = {D0 , D2 , D4 }.

The inverse matrix is

⎛

⎞ −1 1 0 1 M−1 = ⎝ 1 0 1⎠ . 2 0 −1 1 Pick the vertex T := (−1, 1, 1) of the graph H (t, 2). The corresponding vector x defined by (2.6) is x := T · M−1 = (1, −1, 1) . The subset ⎫ ⎧ ⎬ ⎨ Q (T , D) = x1 D0 , x2 D1 , xt Dt−1 = { D0 , D2 , D4 } ⊂ V(D) ⎩          ⎭ D0

D4

D2

of the vertex set of the cycle D is a linearly independent subset (a basis, for our particular vertex T under consideration) of the space Rt such that  T : = (−1, 1, 1) = Q = D0 + D2 + D4 Q∈ Q (T , D)

: = (−1, 1, −1) + (1, 1, 1) + (−1, −1, 1) .

2.1.5 Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle: A Poset-Theoretic Interpretation Pick a vertex T ∈ {1, −1}t , and regard the vertex set V(D) of a symmetric cycle D in the hypercube graph H (t, 2) as a subposet of

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.4 (a) This hypercube graph H ((t := 3), 2) is in fact the Hasse diagram of the tope poset (based at a tope T , and isomorphic to the Boolean lattice of rank t) of the oriented matroid H := (E t , {1, −1}t ). (b) A symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the graph H (t, 2), whose vertex set V(D) is regarded as a subposet of the tope poset of the oriented matroid H, based at T . The inclusion-minimal subset of vertices in the set V(D), such that their sum is the vertex T , is the set Q (T , D) = min V(D) = {D0 , D2 , D4 } of minimal elements of the poset V(D).

the tope poset of the oriented matroid H := (E t , {1, −1}t ), based at the tope T . According to Theorem 11.1 of Ref. [5], the unique inclusion-minimal subset Q (T , D) ⊂ V(D) with the property (2.7) is the set Q (T , D) = min V(D) of minimal elements of the subposet V(D). In the particular case, where the tope poset of the oriented matroid H is based at the positive vertex T(+) , the set Q (T(+) , D) = min V(D) = max+ (V(D)) is a critical committee for H. Example 2.5. Suppose t := 3. Pick the vertex T := (−1, 1, 1) of the hypercube graph H (t, 2), and consider again the symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the graph H (t, 2), depicted in Figure 2.3. Let us transform the graph H (t, 2) in the way such that it becomes the Hasse diagram of the tope poset of the oriented matroid H := (E t , {1, −1}t ), based at the tope T ; see Figure 2.4(a). The unique inclusion-minimal subset Q (T , D) ⊂ V(D) with the property (2.7), detected in Subsection 2.1.4 by means of elementary

15

16 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

linear algebra, is now seen in Figure 2.4(b) as the set min V(D) = {D0 , D2 , D4 } of minimal elements of the subposet V(D), where D0 := (−1, 1 − 1), D2 := (1, 1, 1), and D4 := (−1, −1, 1). Since the poset rank of a vertex in the tope poset is essentially its Hamming distance from the bottom vertex T , we see that the set Q (T , D) is precisely the subset of vertices in the set V(D) that are locally nearest to the vertex T . That is, if (T k1 , T k2 , T k3 ) is a 2path in the cycle D, where T k1 =  T k3 , then we have T k2 ∈ Q (T , D) ⇐⇒ d(T , T k1 ) = d(T , T k3 ) = d(T , T k2 ) + 1 .

2.1.6 Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle: Decompositions by Means of the Tuples With Inclusion-Maximal Positive Parts in the Set of Reoriented Vertices of the Cycle If T ∈ {1, −1}t is a vertex of the hypercube graph H (t, 2), then the inclusion-minimal subset Q (T , D) ⊂ V(D), with the property (2.7), is the set    (2.9) Q (T , D) = − n(T ) max+ − n(T ) V(D) ; see Corollary 11.2 in Ref. [5]. Here − n(T ) V( D ) is the sequence of tuples obtained from the vertex sequence V(D) of the cycle D by reorientation on the negative part n(T ) of the vertex T ; max+ (·) is the subset of all tuples from the resulting sequence that have inclusion-maximal positive parts, and the outermost operation − n(T ) (·) means the reverse reorientation of the tuples with inclusion-maximal positive parts on the negative part n(T ) of the vertex T . Example 2.6. Suppose t := 3. We would like to find the decomposition of the vertex T := (−1, 1, 1) of the hypercube graph H (t, 2) with respect to its symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) depicted in Figure 2.5(a). First, we reorient the vertex T , as well as the vertex set V(D) of the cycle D, on the negative part n(T ) = {1} of the vertex T ;

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.5 (a) A symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H ((t := 3), 2), and a vertex T of the graph H (t, 2). (b) The reorientation of the vertex T on its negative part n(T ) turns T into the positive sign tuple T(+) . The set max+ (− n(T ) V(D)) of tuples with inclusion-maximal positive parts in the reoriented symmetric cycle is the set {− n(T ) D0 , − n(T ) D2 , − n(T ) D4 }. (c) The inclusion-minimal subset of vertices Q (T , D) ⊂ V(D), such that their sum is the vertex T, is the reoriented set 0 2 4 0 2 4 − n(T ) {− n(T ) D , − n(T ) D , − n(T ) D }, that is, Q (T , D) = {D , D , D }.

see Figure 2.5(b). As a result, T turns into the positive tuple T(+) . The set max+ ({− n(T ) D : D ∈ V(D)}) = {− n(T ) D0 , − n(T ) D2 , − n(T ) D4 }, where D0 := (−1, 1 − 1), D2 := (1, 1, 1), and D4 := (−1, −1, 1), is the unique inclusion-minimal subset of tuples in the cycle − n(T ) D such that their sum is T(+) , as explained in Subsection 2.1.6. We see in Figure 2.5(c) that the subset Q (T , D) ⊂ V(D), built up in (2.9), is the set − n(T ) {− n(T ) D0 , − n(T ) D2 , − n(T ) D4 }, that is, Q (T , D) = {D0 , D2 , D4 }.

17

18 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

2.1.7 The Sizes of Vertex Decompositions: Distance Signals Let D := (D0 , D1 , . . . , D2t−1 , D0 ) be a symmetric cycle in the hypercube graph H (t, 2), and let T ∈ {1, −1}t be a vertex of the graph H (t, 2). According to Definition 12.2 of Ref. [5], the (row) distance vector of the cycle D with respect to the vertex T is the sequence    D) zT , D := (zT , D (0), . . . , zT , D (2t − 1)) := d(T , D) : D ∈ V( of the graph distances between T and the vertices of D. We denote by W the 2t × 2t Fourier matrix whose rows and columns are indexed starting with zero. Recall that the (m, n)th entry of the matrix W by definition is e−πımn/ t , 0 ≤ m, n ≤ 2t−1. The Discrete Fourier Transform zTˆ , D of the distance vector zT , D is defined to be the vector zTˆ , D := zT , D · W . Consider the unique inclusion-minimal subset Q (T , D) of the  vertex set V(D) of the cycle D, such that T = Q∈Q (T , D) Q. It is shown in Chapter 12 of Ref. [5] that q(T , D) := |Q (T , D)| = t − that is, q(T , D) = t −

1 2t

2t−1 πk 1  |zTˆ , D (k)|2 · sin2 , 4t k=0 t



|zTˆ , D (k)|2 · sin2

1≤k≤t−1, k odd

πk , t

since

⎧ ⎪ t2 , ⎪ ⎪ ⎪ ⎨0 , zTˆ , D (k) =  ⎪2 −t · (1 − e−πık/t )−1 ⎪ ⎪ ⎪   ⎩ −πıkj /t , + t−1 j =0 zT , D ( j )e

if k = 0 , if k is even, k = 0 , if k is odd ;

see Proposition 12.3 in Ref. [5]. Now let aT , D := (aT , D (0), . . . , aT , D (2t − 1)) denote the autocorrelation of the distance vector zT , D , defined by aT , D (m) :=

2t−1  n=0

zT , D (n) · zT , D ((n + m)

mod 2t) , 0 ≤ m ≤ 2t − 1 .

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.6 Two symmetric cycles D := (D0 , D1 , . . . , D2t−1 , D0 ) and D  := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H ((t := 3), 2). The vertex sets of the cycles are the same, that is, V(D) = V(D  ), but the names of identical vertices in the cycles do not coincide. In other words, the vertex    ). For example, we sequences of the cycles are different: V(D) = V(D  4 1   have V(D)  D := (−1, −1, 1) =: D ∈ V(D ).

By Eq. (13.5) of Ref. [5] we have q(T , D) = t −

 1 aT , D (0) − aT , D (2) . 4

Example 2.7. (i) Suppose t := 3. Consider symmetric cycles D := (D0 , D1 , . . . , D2t−1 , D0 ) and D  := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H (t, 2), depicted in Figures 2.6(a) and (b), respectively. The vertex sets V(D) and V(D  ) of the cycles are the same, that is, V(D) = V(D  ), but the names of identical vertices  D) in the cycles do not coincide. Thus, the vertex sequences V(    and V( D ) of the cycles differ. For example, we have V( D)  D4 :=  D  ). (−1, −1, 1) =: D1 ∈ V( According to Definition 12.2 of Ref. [5], the distance vectors zT , D and zT , D  of the cycles D and D  with respect to the vertex T := (−1, 1, 1) are ⎞ ⎛ 1 3 2t−1 zT , D

⎜ =⎝ 1 , ↑

⎛

d(T , D )

2 ,

↑

d(T , D0 )

1 , ↑

d (T , D1 )

2 ,

↓

1 , ↑

↑

d(T , D2 )

2 d(T , D2t−1 )

↓

2 ,

)

d(T , D4 ) d(T , D3 )

↓

1 ,

d(T , D

↓

d(T , D2 )

d(T , D0 )

⎜ zT , D  = ⎝ 2 ,

d(T , D )

↓

1 ,

↓

2 , ↑

d(T , D4 )

1

⎟ ⎠ , ⎞ ⎟ ⎠ .

19

20 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

Figure 2.7 A symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H ((t := 6), 2), and a vertex T ∈ {1, −1}t of the graph. The subset set Q (T , D) := {Q0 , Q1 , Q2 , Q3 , Q4 } is the unique inclusion-minimal  of the vertex set V(D) of the cycle D, such that T = Q∈Q (T , D) Q.

We have aT , D (0) : =

2t−1 

zT , D (n) 2 = 12 + 22 + 12 + 22 + 12 + 22 = 15 ,    n=0 d(T , Dn )

aT , D (2) : =

2t−1  n=0

zT , D (n) · zT , D ((n + 2) mod 2t)       d (T , Dn )

d (T , D(n+2)

mod 2t )

= 1 · 1 + 2 · 2 + 1 · 1 + 2 · 2 + 1 · 1 + 2 · 2 = 15 . We obtain q(T , D) = t −

 1 1 aT , D (0) − aT , D (2) = 3 − (15 − 15) = 3 , 4 4

and we certainly have q(T , D  ) = q(T , D) = 3. (ii) Suppose t : = 6. Given a symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H (t, 2), depicted in Figure 2.7, the distance vector of the cycle D with respect to the vertex T := (−1, 1, 1, 1, 1, −1) is ⎛ ⎞ 3

1 ↓

5

↓

7

↓

2t−1

9

↓

↓

↓

zT , D = ⎝ 4, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4 , 3 ⎠ . ↑

0

↑

2

↑

4

↑

6

↑

8

↑

10

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

We have aT , D (0) : =

2t−1 

zT , D (n) 2    n=0 d (T , Dn )

= 4 + 32 + 22 + 32 + 22 + 32 + 22 + 32 + 42 + 32 2

+ 42 + 32 = 114 , 2t−1  aT , D (2) : = zT , D (n) · zT , D ((n + 2) mod 2t)       n=0 d (T , D(n+2)

d(T , Dn )

mod 2t )

= 4·2+3·3+2·2+3·3+2·2+3·3 + 2 · 4 + 3 · 3 + 4 · 4 + 3 · 3 + 4 · 4 + 3 · 3 = 110 . Thus, we have  1 1 aT , D (0) − aT , D (2) = 6 − (114 − 110) = 5 . 4 4 Indeed, this is the cardinality of the vertex subset Q (T , D) = {Q0 , Q1 , . . . , Q4 } ⊂ V(D); see Figure 2.7. q(T , D) = t −

2.1.8 Basic Metric Properties of Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle In view of Remark 13.2 of Ref. [5], for any vertex T ∈ {1, −1}t of the hypercube graph H (t, 2), and for any symmetric cycle D in the graph H (t, 2), we have  1 (2.10) d(T , Q) = (q(T ) − 1)t , 2 Q∈Q (T , D) and q(T ) = 1 +

2 t



d(T , Q) .

(2.11)

Q∈Q (T , D)

As noted in Remark 13.3 of Ref. [5], if T ∈ V(D), then we have   1 q(T )2 − 1 t , (2.12) d(Qi , Q j ) = 4 0≤i < j ≤q(T )−1  4 (2.13) q(T ) = 1 + d(Qi , Q j ) , i< j t

21

22 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

and 

d(Qi , Q j ) =

0≤i < j ≤q(T )−1

  1 q(T ) + 1 d(T , Q) . 2 Q∈ Q (T , D)

(2.14)

Example 2.8. For a symmetric cycle D of the hypercube graph H ((t := 6), 2), and for the vertex T := (−1, 1, 1, 1, 1, −1) ∈ {1, −1}t of the graph H (t, 2), depicted in Figure 2.7, we have   d(T , Q) d(T , Q) = Q∈{Q0 , Q1 , ..., Q4 }

Q∈Q (T , D)

= d(T , D2 ) + d(T , D4 ) + d(T , D6 )          d(T , Q0 )

d(T , Q1 )

d(T , Q2 )

+ d(T , D9 ) + d (T , D2t−1 )       d(T , Q4 )

d(T , Q3 )

= 2 + 2 + 2 + 3 + 3 = 12 1 1 = (q(T ) − 1)t = (5 − 1)6 , 2 2 and q(T ) = 1 + Further,  0≤i < j ≤q(T )−1

2 t



d(T , Q) = 1 +

Q∈Q (T , D)

2 · 12 = 5 . 6

d ( Qi , Q j ) = d (Q0 , Q1 ) + d(Q0 , Q2 ) + d(Q0 , Q3 )          4

2

5

+ d(Q , Q ) + d(Q1 , Q2 ) + d( Q1 , Q3 )          0

4

3

2

5

+ d(Q1 , Q4 ) + d(Q2 , Q3 )       5 2

3 3

+ d(Q , Q ) + d(Q , Q4 ) = 36       4

2

5

  1 2 1 5 −1 6, = q(T )2 − 1 t = 4 4 and

 q(T ) =

4 1+ d (Qi , Q j ) = i< j t

 1+

4 · 36 = 5 , 6

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

and



d(Qi , Q j ) = 36

0≤i < j ≤q(T )−1

=

  1 1 q(T ) + 1 d(T , Q) = (5 + 1)12 . 2 2 Q∈ Q (T , D)

2.1.9 The Negative Parts of Vertices, Graph Distances, and the Scalar Products of Vertices Let D be a symmetric cycle in a hypercube graph H (t, 2). Looking at relations (2.10) and (2.11) for vertices T ∈ {1, −1}t of the graph H (t, 2), note that the following implications hold: e ∈ E t , T (e) = −1 =⇒

|{Q ∈ Q (T , D) : Q(e) = −1}| =

q(T ) 2

We see that   q(T )

  q(T ) | n(Q)| = + 2 2 Q∈ Q (T , D) e∈n(T ) e∈p(T )

 q(T ) q(T ) = | n(T )| · + (t − | n(T )|) · 2 2

.

.

Remark 2.9. For any vertex T ∈ {1, −1}t of the hypercube graph H (t, 2), and for any symmetric cycle D in the graph H (t, 2), we have (i) 

| n(Q)| = | n(T )| +

Q∈ Q (T , D)

and q(T ) = 1 − (ii)

 Q∈Q (T , D)

2 2 | n(T )| + t t

 1 q(T ) − 1 t , 2



| n(Q)| .

(2.15)

Q∈ Q (T , D)

d(T , Q) = −| n(T )| +

 Q∈ Q (T , D)

| n( Q)| .

(2.16)

23

24 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

Example 2.10. Suppose t := 6. For a symmetric cycle D of the hypercube graph H (t, 2), and for the vertex T := (−1, 1, 1, 1, 1, −1) ∈ {1, −1}t of the graph H (t, 2), depicted in Figure 2.7, we have   | n(Q)| | n(Q)| = Q∈ Q (T , D)

Q∈{Q0 , Q1 , ..., Q4 }

= | n(D2 )| + | n(D4 )| + | n( D6 )| + | n(D9 )|             2

4

2

3

+ | n(D2t−1 )|    3

= 14 = | n(T )| + = 2+

 1 5−1 6, 2

and q(T ) = 5 = 1 −

 1 q(T ) − 1 t 2

2 2 | n(T )| + t t



| n(Q)| = 1 −

Q∈ Q (T , D)

Further, we have  d(T , Q) = 12 = −| n(T )| + Q∈Q (T , D)



2 2 · 2 + · 14 . 6 6

| n(Q)| = −2 + 14 .

Q∈Q (T , D)

Let us now turn to relations (2.12)–(2.14). Let T ∈ {1, −1}t be a vertex of the hypercube graph H (t, 2) with its symmetric cycle D, such that T ∈ V(D ). Relations (2.16) and (2.14) yield    1 q(T ) + 1 d(Qi , Q j ) = d(T , Q) 2 0≤i < j ≤q(T )−1 Q∈ Q (T , D) "  ! 1 q(T ) + 1 −| n(T )| + | n(Q)| . = 2 Q∈ Q (T , D) Using (2.15), we obtain   1 q(T ) + 1 d(Qi , Q j ) = 2 0≤i < j ≤q(T )−1 ! × −| n(T )| +

"



| n(Q)|

Q∈ Q (T , D)

1! 2 2 = 1 − | n(T )| + 2 t t !  × −| n(T )| +



Q∈Q (T , D)

Q∈ Q (T , D )

| n(Q)| + 1

" | n(Q)| .

"

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Remark 2.11. Let D be a symmetric cycle in a hypercube graph H (t, 2). Let T ∈ {1, −1}t be a vertex of the graph H (t, 2), such that T ∈ V(D). We have ! " 1 1  d(Qi , Q j ) = 1 − | n(T )| + | n(Q)| t t Q∈Q (T , D) 0≤i < j ≤q(T )−1 ! "  × −| n(T )| + | n(Q)| . 

Q∈Q (T , D)

Example 2.12. Suppose t := 6. For the vertex T := (−1, 1, 1, 1, 1, −1) ∈ {1, −1}t of the hypercube graph H (t, 2) with its symmetric cycle D, depicted in Figure 2.7, we have 

d(Qi , Q j ) = 36

0≤i < j ≤q(T )−1

! " 1 1  = 1 − | n(T )| + | n(Q)| t t Q∈Q (T , D) " !  | n(Q)| × −| n(T )| + !

= 1−

Q∈Q (T , D)

" ! " 1 1 · 2 + · 14 · −2 + 14 . 6 6

For convenience, we now reformulate relations (2.12)–(2.14) and Remark 2.11 via the scalar products of vertices: Remark 2.13. Let D be a symmetric cycle in a hypercube graph H (t, 2). Let T ∈ {1, −1}t be a vertex of the graph H (t, 2), such that T ∈ V(D). We have: (i)  0≤i < j ≤q(T )−1

Qi , Q j  =

 1 1 − q(T ) t , 2

q(T ) = 1 −

 2  Qi , Q j  . t 0≤i < j ≤q(T )−1

25

26 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

(ii) 

Qi , Q j  =

0≤i < j ≤q(T )−1

q(T ) 2

t

! " 1 1  − 2 1 − | n(T )| + | n(Q)| t t Q∈ Q (T , D) ! "  × −| n(T )| + | n(Q)| . Q∈ Q (T , D)

Example 2.14. Suppose t := 6. For the vertex T := (−1, 1, 1, 1, 1, −1) ∈ {1, −1}t of the hypercube graph H (t, 2) with its symmetric cycle D, depicted in Figure 2.7, we have  0≤i < j ≤q(T )−1

Qi , Q j  = Q0 , Q1  + Q0 , Q2  + Q0 , Q3           2

−2 1

−4 1

+  Q , Q  +  Q , Q  +  Q , Q3  + Q1 , Q4              0

4

0

2

2

−4 4 3

−4

+  Q , Q  + Q , Q  + Q , Q  = −12          2

3

0

2

−4

4

2

  1 1 = 1 − q(T ) t = 1−5 6, 2 2 and q(T ) = 5 = 1 −

 2 2 Qi , Q j  = 1 − (−12) . t 0≤i < j ≤q(T )−1 6

We also have 

 Qi , Q j  = −12

0≤i < j ≤q(T )−1

=

! " 1 1  t − − | T | + 2 1 n( ) n( ) | | Q 2 t t Q∈ Q (T , D) ! "  × −| n(T )| + | n(Q)| q(T )

Q∈ Q (T , D)

=

5 2



6−2 1−

   1 1 · 2 + · 14 · −2 + 14 . 6 6

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Now, if T  and T  are two vertices of a hypercube graph H (t, 2), then we have #   $ , t − 2d(T  , T  ) = T  , T   = =

Q ∈Q (T  , D) Q ∈Q (T  , D)





 Q , Q  =

Q ∈Q (T  , D) , Q ∈Q (T  , D)



t − 2d(Q , Q )



Q ∈Q (T  , D) , Q ∈Q (T  , D)



= q(T  )q(T  )t − 2

d(Q , Q ) .

Q ∈ Q (T  , D) , Q ∈Q (T  , D)

Remark 2.15. If T  , T  ∈ {1, −1}t are vertices of the hypercube graph H (t, 2) with its symmetric cycle D, then we have −d(T  , T  ) +



d(Q , Q ) =

Q ∈ Q (T  , D) , Q ∈Q (T  , D)

 1 q(T  )q(T  ) − 1 t . 2

Example 2.16. Suppose t := 6. Pick the vertices T  := (−1, 1, 1, 1, 1, −1) ∈ {1, −1}t and T  := (1, −1, −1, 1, 1, 1) of the hypercube graph H (t, 2) with its symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ), depicted in Figure 2.8. We have  d(Q , Q ) −d(T  , T  ) + Q ∈Q (T  , D) , Q ∈Q (T  , D)

= −4 + d(Q0 , Q0 ) + d(Q0 , Q 1 ) + d(Q0 , Q2 )          1

3

6

+ d (Q 1 , Q0 ) + d( Q1 , Q1 ) + d(Q 1 , Q 2 )          3

4

1

+ d(Q2 , Q0 ) + d(Q2 , Q1 ) + d(Q 2 , Q 2 )          5 3

0

1 3

1

2 3

4 4

0

4 4

1

1 4

+ d(Q , Q ) + d(Q , Q ) + d(Q , Q2 )          + d(Q , Q ) + d(Q , Q ) + d(Q , Q2 )          2

6

3

  1 1 = 42 = q(T  )q(T  ) − 1 t = 5·3−1 6. 2 2

27

28 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

Figure 2.8 Two instances of the same symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H ((t := 6), 2), and two vertices T  and T  of the graph. 4 } is the unique inclusion(a) The set Q (T  , D) := {Q0 , Q1 , Q2 , Q3 , Q  minimal subset of the set V(D) such that T = Q ∈Q (T  , D) Q . 2 (b) The set Q (T  , D) := {Q0 , Q1 , Q } is the unique inclusion-minimal subset of the set V(D) such that T  = Q ∈Q (T  , D) Q .

2.1.10 A Basic Statistic on Vertex Decompositions in a Hypercube Graph with Respect to a Symmetric Cycle For an arbitrary symmetric cycle D in a hypercube graph H (t, 2), in view of Eq. (13.7) of Ref. [5], we have  |Q (T , D)| = 2t−1 t . (2.17)    t T ∈{1, −1}

q(T )

Recall that this quantity coincides with the number of edges of the graph H (t, 2).

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

By Theorem 13.6 of Ref. [5], for any odd integer , 1 ≤  ≤ t, we have  (2.18) c (t) := |{T ∈ {1, −1}t : q(T ) = }| = 2 t . Note that if t is even, then for any odd integer , 1 ≤  < t, we have c (t) = ct− (t) . If t is odd, then for any odd integer , 1 ≤  ≤ t, we have c (t) = (1 + t − )c1+t− (t) . Example 2.17. Suppose t := 3. For a symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H (t, 2), depicted in Figure 2.4(a), we have  q(T ) = q(T ) + q(−T )       T ∈{1, −1}t 3 3  q(D) = 12 = 2t−1 t = 23−1 3 . +   D∈V( D )

1

We also have

t

|{T ∈ {1, −1}t : q(T ) = 1}| = 2 and

1

t

|{T ∈ {1, −1}t : q(T ) = 3}| = 2

3

 := 2 31 = 6 ,  := 2 33 = 2 .

2.1.11 Equinumerous Decompositions of a Vertex in a Hypercube Graph with Respect to Its Symmetric Cycles In this subsection, we will count the number of symmetric cycles D in a hypercube graph H (t, 2) that provide equinumerous decompositions Q (T , D) of an arbitrary vertex T ∈ {1, −1}t . It is easy to see that each of the 2t vertices of the graph H (t, 2) belongs to 12 t! symmetric cycles. Indeed, the graph H (t, 2) regarded as the Hasse diagram of the tope poset of the oriented matroid H := (E t , {1, −1}t ) can be based at any vertex B ∈ {1, −1}t , and such a poset, with its least element B contained in t! maximal chains, is a principal order ideal of a binomial poset whose factorial function is n!; see Example 3.18.3b in Ref. [7]. It now suffices to note that any

29

30 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

symmetric cycle of H (t, 2) containing B as its vertex is a union of two maximal chains of the tope poset based at B. Thus, there are 2t1 · 2t · 12 t! symmetric cycles in H (t, 2): #{D : D symmetric cycle of H (t, 2)} = 2t−2 (t − 1)! . See Sequence A002866 in online Ref. [8] on the corresponding integer sequence. Recall that for any symmetric cycle D of the hypercube graph H (t, 2), by (2.17) and (2.18) we have   t  q(T ) = 2t−1 t = 2   . 1≤≤t: j odd

T ∈{1, −1}t

As a consequence, we have  

q(T ) = 2t−1 (t − 1)!

1≤≤t:  odd

D: T ∈{1, −1}t D symmetric cycle of H (t, 2)

Since

 D: D symmetric cycle of H (t, 2)



q(T ) =

  t .

 t  1 t−1 2 (t   − 1)! 2t 1≤≤t:  odd

(t − 1)!   t  =   , 2 1≤≤t:  odd

for any vertex T ∈ {1, −1}t of the graph H (t, 2), we come to the following result: Proposition 2.18. For any vertex T ∈ {1, −1}t of the hypercube graph H (t, 2), and for an odd integer , 1 ≤  ≤ t, we have (t − 1)!  t  . #{D : D symmetric cycle of H (t, 2), q(T , D) = } =  2 Example 2.19. Suppose t := 3, and consider the vertex T := (−1, 1, 1) of the hypercube graph H (t, 2), depicted in Figure 2.9(a). There are 2t−2 (t − 1)! = 23−2 (3 − 1)! = 4 symmetric cycles in the graph H (t, 2). We have #{D : D symm. cycle of H (t, 2), q(T , D) = 1} (t − 1)!  t  (3 − 1)! 3 = = =3; 1 1 2 2

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.9 (a) The hypercube graph H ((t := 3), 2), and its vertex T . (b)(c)(d) The three symmetric cycles D of the graph H (t, 2) such that q(T , D) = 1. (e) The unique symmetric cycle D of the graph H (t, 2) such that q(T , D) = 3. Here Q (T , D) = {D0 , D2 , D4 }.

see Figures 2.9(b), (c) and (d); and we have #{D : D symm. cycle of H (t, 2), q(T , D) = 3} (t − 1)!  t  (3 − 1)! 3 = =1; = 3 3 2 2 see Figure 2.9(e).

31

32 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

2.1.12 Change of Cycle, Change of Basis If we are interested in the decompositions of a vertex T of a hypercube graph H (t, 2) with respect to its symmetric cycles D  and D  , then changing cycles is essentially the same as changing from one (maximal positive) basis of the space Rt to another. Therefore, we have x(T , D  ) = x(T , D  ) · M(D  )M(D  )−1 . Example 2.20. Suppose t := 6, and pick the vertex T := (1, −1, −1, 1, 1, 1) of the hypercube graph H (t, 2). We associate with a symmetric cycle R := (R 0 , R 1 , . . . , R 2t−1 , 0 R ) of the graph H (t, 2), depicted in Figure 2.10(a), the matrix ⎛ ⎞ 1 1 1 1 1 1 ⎜−1 1 1 1 1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜−1 −1 1 1 1 1⎟ M(R) := ⎜ ⎟ , ⎜−1 −1 −1 1 1 1⎟ ⎜ ⎟ ⎝−1 −1 −1 −1 1 1⎠ −1 −1 −1 −1 −1 1 and we associate with a symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ), depicted in Figure 2.10(b), the matrices ⎛ ⎞ 1 1 1 −1 −1 1 ⎜1 1 1 1 −1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 1 1 1 −1 −1⎟ M(D) : = ⎜ ⎟, ⎜1 −1 1 1 −1 −1⎟ ⎜ ⎟ ⎝1 −1 1 1 1 −1⎠ 1 −1 −1 1 1 −1 and

⎛

M(D)−1

1 ⎜ 0 ⎜ 1⎜ ⎜ 0 = ⎜ 2 ⎜−1 ⎜ ⎝ 0 0

0 0 0 0 1 −1 0 0 0 1 0 0 0 0 −1 1 −1 0

⎞ 0 1 0 0⎟ ⎟ ⎟ 1 −1⎟ ⎟. 0 0⎟ ⎟ 1 0⎠ 0 0

The unique inclusion-minimal subset Q (T , R) of the vertex set V(R),  such that T = Q∈ Q (T , R) Q, is the set {R 0 , R 3 , R 7 }, that is, x(T , R) = (1, −1,

0,

1,

0,

0) .

Symmetric Cycles in Hypercube Graphs, and Vertex Decompositions

Figure 2.10 A vertex T of the hypercube graph H ((t := 6), 2), and two symmetric cycles R and D in the graph. 3 , R 7 } is the unique inclusion-minimal subset (a) The set Q (T , R) = {R 0 , R  of the set V(R) such that T = Q∈Q (T , R) Q. 5 (b) The set Q (T , D) := {D1 , D , D8 } is the unique inclusion-minimal subset of the set V(D) such that T = Q∈Q(T , D) Q.

In order to find the decomposition of the vertex T with respect to the symmetric cycle D, we use the formula x(T , D) = x(T , R) · M(R)M(D)−1 , and we see that x(T , D) = (0,

1, −1,

that is, Q (T , D) = {D1 , D5 , D8 }.

0,

0,

1) ,

33

34 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

2.1.13 Circular Translations of Vertex Decompositions Remark 2.21. Let T ∈ {1, −1}t be a vertex of the hypercube graph H (t, 2), and let D := (D0 , D1 , . . . , D2t−1 , D0 ) be a symmetric cycle in H (t, 2). Assume that  D) ⊃ Q (T , D) (D0 , D1 , . . . , D2t−1 ) = : V( = (Di 0 , Di 1 , . . . , Di q(T )−1 ) , for some indices i 0 < i 1 < · · · < q(T ) − 1. For any s ∈ Z, we have  D(i j +s) mod 2t ∈ {1, −1}t . 0≤ j ≤q(T )−1

Example 2.22. Suppose t := 6, and pick the vertex T := (1, −1, −1, 1, 1, 1) of the hypercube graph H (t, 2). For a symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the graph H (t, 2), depicted in Figure 2.11, we have Q (T , D) = {D1 , D5 , D8 }.  Now suppose s := 2, and note that the sum Dk ∈ Q (T , D) Dk+s := D1+2 + D5+2 + D8+2 = D3 + D7 + D10 = T(−) is also a vertex of the graph H (t, 2).

2.2 Coherent Vertex Decompositions in  (t, 2) and H (t, 2) Hypercube Graphs H  (t, 2) denote the hypercube graph on its vertex set {0, 1}t . For Let H vertices T  and T  , the pair {T  , T  } by definition is an edge of  (t, 2) if and only if the Hamming distance |{e ∈ E t : T  (e) = T  (e)}| H between the tuples T  and T  is 1. Following Section 1 of Ref. [9], we recall that a family of  (t, 2) includes, interesting graphs related to the hypercube graph H in particular, Fibonacci cubes (see Refs. [10, 11, 12]), Lucas cubes (see Ref. [13]), generalized Fibonacci cubes (see Ref. [14]), kFibonacci cubes (see Ref. [15]), Fibonacci-run graphs (see Refs. [9, 16]) and daisy cubes (see Ref. [17]).  in the hypercube graph H  (t, 2) to We define a symmetric cycle D be a 2t-cycle, with its vertex sequence  1, . . . , D  2t−1 ) ,  0, D  ) := (D  D V(

Coherent Decompositions in Hypercube Graphs with Vertex Sets {0, 1}t and {1, −1}t

Figure 2.11 Two instances of the same symmetric cycle D := (D0 , D1 , . . . , D2t−1 , D0 ) in the hypercube graph H ((t := 6), 2). (a) Pick the vertex T := (1, −1, −1, 1, 1, 1) ∈ {1, −1}t of the graph H (t, 2). , D8 } is the unique inclusion-minimal subset of The set Q (T , D) = {D1 , D5 the set V(D) such that T = Q∈Q (T , R) Q.  (b) Suppose s := 2. The sum Dk ∈Q (T , R) Dk+s = D1+2 + D5+2 + D8+2 = D3 + D7 + D10 is the vertex T(−) of the graph H (t, 2).

and with its vertex set  ) := {D  0, D  1, . . . , D  2t−1 } , V(D such that  k+t := T(+) − D k , 0 ≤ k ≤ t − 1 ; D cf. definition (2.1)–(2.3).  (t, 2) Coherent decompositions of vertices in hypercube graphs H and H (t, 2) are implemented by means of the standard conversions {0, 1} → {1, −1} :

x → 1 − 2x ,

0 → 1 ,

1 → −1 ,

35

36 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

and 1 (1 − z) , 2 whose derived bijective maps are {1, −1} → {0, 1} :

z →

{0, 1}t → {1, −1}t :

1 → 0 ,

−1 → 1 ,

T → T(+) − 2T ,

(2.19)

and 1 (+) (T − T ) . (2.20) 2 Given a symmetric cycle D in the hypercube graph H (t, 2) on its vertex set {1, −1}t , let us regard bijection (2.20) as the (de)composition {1, −1}t → {0, 1}t :

T →

(2.20)



T =

{1, −1}t −−−→ {0, 1}t : Q

Q∈ Q (T , D)

"  1 ! (+) Q T − 2 Q∈ Q (T , D) "    1! Q = q(T , D)T(+) − q(T , D) − 1 T(+) − 2 Q∈ Q (T , D)

→

= −

 1   1 T(+) − Q . q(T , D) − 1 T(+) + 2 2 Q∈Q (T , D)

 in the hypercube graph H  (t, Remark 2.23. For a symmetric cycle D t  (t, 2), there 2) on its vertex set {0, 1} , and for any vertex T of H  (T , D  ) ⊂ V(D  ), of odd exists a unique inclusion-minimal subset Q cardinality, such that   1    . (T , D )| − 1 T(+) + T = − |Q Q 2  , D):    (T Q∈Q  Q=(0, ..., 0)

Example 2.24. Suppose t := 6. Figure 2.12 depicts two coherent  in the hypercube graphs H (t, 2) symmetric cycles D and D   ) of the and H (t, 2), respectively. The vertex sets V(D) and V(D cycles are in one-to-one correspondences (2.20) and (2.19) with each other.

Coherent Decompositions in Hypercube Graphs with Vertex Sets {0, 1}t and {1, −1}t

 in the hypercube Figure 2.12 Coherent symmetric cycles D and D  graphs H ((t := 6), 2) and H (t, 2). The bijections between the vertex sets  of the cycles are given in (2.20) and (2.19). V(D) and V( D) (a) The set Q (T , D) := {Q0 , Q1 , . . . , Q4 } is the unique inclusion-minimal  subset of the vertex set V(D) of the cycle D, such that T = Q∈Q (T , D) Q.  (T , D)  := {  Q1 , . . . ,  Q4 } is the unique inclusion-minimal (b) The set Q Q0 ,    (T , D)|  −   such that T = − 1 | Q subset of the vertex set V( D) of the cycle D, 2  (+)   1 T + Q∈ Q (T , D)  Q.

 (t, For the vertex T := (1, 0, 0, 0, 0, 1) ∈ {0, 1}t of the graph H 2), the set ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ 0 ,   (T , D ) : = 2 , Q 3 , Q 4 Q Q Q1 , Q ⎪     ⎪ ⎩ ⎭  D2

 D4

 D6

 D9

 D2t−1

= {(0, 0, 0, 0, 1, 1), (0, 1, 0, 0, 0, 1), (1, 1, 1, 0, 0, 1), (1, 0, 1, 1, 0, 0), (1, 0, 0, 1, 1, 0)}

37

38 A 2D Perspective on Discrete Hypercubes and the Power Sets of Finite Sets

is the unique inclusion-minimal subset, of odd cardinality, of the  ) such that vertex set V(D   1    . (T , D )| − 1 T(+) + T = − |Q Q 2   (T , D ) Q∈Q

Indeed, we have 1 T := (1, 0, 0, 0, 0, 1) = − (5−1)·(1, 1, 1, 1, 1, 1)+(3, 2, 2, 2, 2, 3) . 2 In practice, it is convenient to translate decomposition problems,  (t, 2) and its symmetric cycles, to the for the hypercube graph H hypercube graph H (t, 2) on its vertex set {1, −1}t , and then to send  (t, 2). their solutions (found more or less easily) back to the graph H t Let X , Y ∈ {1, −1} be two vertices of the hypercube  , Y ∈ {0, 1}t be the graph H (t, 2), where t is even, and let X  (t, 2), namely, X  := corresponding vertices of the hypercube graph H 1 1 (+) (+) (+)    (T − X ) and Y := 2 (T − Y ). Let hwt(T ) := T , T  denote 2 the Hamming weight of a vertex T ∈ {0, 1}t , that is, the number of 1’s in T . We have    , Y  = 2(hwt(X ) + hwt(Y )) − t , X , Y  = 0 ⇐⇒ X 4 and if 4|t (i.e., t is divisible by 4), then we have | n(X )| = | n(Y )| =: s , X , Y  = 0  ) = hwt(Y ) =: s , X  , Y  = s − t . ⇐⇒ hwt(X 4

Chapter 3

Vertex Decompositions in Hypercube Graphs, and Dehn–Sommerville Type Relations

An arrangement of distinct oriented lines crossing the origin (0, 0) of the two-dimensional Euclidean space R2 on a piece of paper represents implicitly a rank 2 system of homogeneous strict linear inequalities. If there is no point lying on the positive sides of all lines of the arrangement, then the inclusion-maximal positive parts of topes of the oriented matroid realized by the arrangement are the multi-indices of maximal feasible subsystems of an infeasible system. If the system has at least five maximal feasible subsystems, then the numbers of feasible subsystems satisfy relations derived from the Dehn–Sommerville equations for the face numbers of the boundary complexes of simplicial convex polytopes. In such a situation, the tope set of our oriented matroid, regarded as the vertex set of a symmetric cycle in a hypercube graph, leads us to Dehn–Sommerville type relations for a certain abstract simplicial complex associated with the decomposition set for the positive vertex of the graph. In this chapter, we present Dehn–Sommerville type relations that are valid for arbitrary vertices of hypercube graphs with sufficiently large decomposition sets.

Symmetric Cycles Andrey O. Matveev c 2023 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4968-81-2 (Hardcover), 978-1-003-43832-8 (eBook) www.jennystanford.com

40 Vertex Decompositions and Dehn–Sommerville Type Relations

In Section 3.1 we recall some enumerative results on rank 2 infeasible systems of linear inequalities related to arrangements of oriented lines in the plane. In Section 3.2 we present Dehn– Sommerville type relations for the numbers of faces of abstract simplicial complexes associated with large-size decomposition sets for vertices of hypercube graphs.

3.1 Dehn–Sommerville Equations for the Feasible Subsystems of Rank 2 Infeasible Systems of Homogeneous Strict Linear Inequalities Let D be a symmetric cycle in the hypercube graph H (t, 2) on its vertex set {1, −1}t , such that V(D)  T(+) , that is, the positive vertex T(+) is not a vertex of the cycle D. As mentioned in Section 2.1.1, the vertex set V(D) of the cycle D is the set of topes of a rank 2 oriented matroid H D := (E t , V(D)). Thus, we can consider the representation of H D by a certain central line arrangement {x ∈ R2 : ae , x = 0, e ∈ E t }

(3.1)

in the plane R , with the set of corresponding normal vectors 2

A := {ae : e ∈ E t } .

(3.2)

If the cardinality of the unique inclusion-minimal subset  Q (T(+) , D) ⊂ V(D), with the property Q∈Q (T(+) , D) Q = T(+) , is not too small, namely q(T(+) ) := q(T(+) , D) := |Q (T(+) , D)| ≥ 5 , then for the set of normal vectors (3.2) we have |{a ∈ A : a ∈ C> }| ≥ 2 ,

(3.3)

for any open half-plane C> ⊂ R2 bounded by a one-dimensional subspace of R2 ; see, e.g., the proof of Proposition 2.33 in Ref. [18]. As noted several times in Section 2.1, we have Q (T(+) , D) = max+ (V(D)) ,

Dehn–Sommerville Equations for Rank 2 Infeasible Systems of Linear Inequalities

that is, the decomposition set Q (T(+) , D) for the positive vertex T(+) with respect to the cycle D is precisely the subset max+ (V(D)) ⊂ V(D) of vertices with inclusion-maximal positive parts. Example 3.1. Suppose t := 6, and consider a symmetric cycle D in the hypercube graph H (t, 2), depicted in Figure 3.1(a). The positive vertex T(+) is not a vertex of the cycle D. The vertex set V(D) of the cycle D is the set of topes of a rank 2 oriented matroid H D . A central arrangement of oriented lines in the plane R2 that realizes H D is depicted in Figure 3.1(b). We denote by A := {a1 , . . . , at } the set of normal vectors defining the lines of the arrangement. Since for the unique inclusion-minimal subset Q (T(+) , D) ⊂  V(D), such that Q∈Q (T(+) , D) Q = T(+) , we have q(T(+) ) ≥ 5, any open half-plane C> bounded by a one-dimensional subspace of the space R2 contains at least two vectors of the set A; see Figure 3.1(c). Denote by ν j the number of feasible subsystems, of cardinality j , of the infeasible system of homogeneous strict linear inequalities   ae , x > 0 : x ∈ R2 , ae ∈ A (3.4) associated with the arrangement (3.1). Since the condition (3.3) is fulfilled, the quantities ν j satisfy the relations (where ‘x’ is a formal variable) ⎧ t ⎪ ⎪ν j = j , if 0 ≤ j ≤ 2 , ⎨ νt−1 = νt = 0 , " !  " ⎪t ! t   ⎪ ⎩ −ν j (x− 1)t− j = − t (−1) j t − ν j xt− j j =3

j =3

j

j

(3.5) called the Dehn–Sommerville equations for the feasible subsystems of the system (3.4); see, e.g., Proposition 3.53 in Ref. [18]. In particular, – if t := 5, then ν3 = – if t := 6, then ν3 = ν4 =

t

t 3t  2

2

−t ;

− 2t + 4 , − 3t + 6 ;

41

42 Vertex Decompositions and Dehn–Sommerville Type Relations

Figure 3.1 (a) A symmetric cycle D in the hypercube graph H ((t := 6), 2). The positive vertex does not belong to the cycle: T(+) ∈ V(D). The subset Q (T(+) , D) = {Q0 , Q1 , . . . , Q4 } of the vertex set V(D) of the cycle D consists of the vertices with inclusion-maximal positive parts. (b) The vertex set V(D) of the cycle D is the set of topes of the rank 2 oriented matroid H D represented by a central line arrangement {L 1 , . . . , L t } in the plane R2 . The regions of the arrangement marked by disks correspond to the vertices of the graph H (t, 2) that have inclusion-maximal positive parts. (c) The set of normal vectors A := {ae : e ∈ E t } defining the oriented lines L e := {x ∈ R2 : ae , x = 0} of the arrangement depicted in Figure (b). Since q(T(+) ) ≥ 5, we have |A ∩ C> | ≥ 2, for any open half-plane C> := {x ∈ R2 : c, x > 0} bounded by a one-dimensional subspace {x ∈ R2 : c, x = 0} ⊂ R2 , where c = 0.

Dehn–Sommerville Type Relations for Vertex Decompositions

– if t := 7, then   ν4 = 2ν3 − 2 4t + 3t ,   ν5 = ν3 − 4t + 2t − t . For any j , where 3 ≤ j ≤ t − 2, by Corollary 3.55(i) of Ref. [18] we have t  j

−ν j = −

j "   !t −νi . (−1)i t−i j −i i

(3.6)

i =3

Recall also that t−2 

(−1) j ν j = 0 .

j =1

3.2 Dehn–Sommerville Type Relations for Vertex Decompositions in Hypercube Graphs. I Given a vertex T ∈ {1, −1}t of the hypercube graph H (t, 2) with its symmetric cycle D, consider the corresponding decomposition  Q T = Q∈ Q (T , D)

of T with respect to the cycle D, for a unique inclusion-minimal subset Q (T , D) ⊂ V(D). Suppose that q(T ) := q(T , D) := |Q (T , D)| ≥ 5 , and consider the oriented matroid YT , D := − n(T ) (H D ) obtained from the rank 2 oriented matroid H D := (E t , V(D)), with its set of topes V(D), by reorientation of H D on the negative part n(T ) of the tope T . Associate with the abstract simplicial complex  := acyclic (YT , D )   := A ⊂ E t : ∃F ∈ − n(T ) V(D) with p(F ) ⊇ A

(3.7)

43

44 Vertex Decompositions and Dehn–Sommerville Type Relations

of acyclic subsets of the ground set of the oriented matroid YT , D its long f -vector   (3.8) f (; t) := f0 (; t), f1 (; t), . . . , ft (; t) ∈ Nt+1 defined by f j (; t) := #{F ∈  : |F | = j } , 0 ≤ j ≤ t .

(3.9)

In view of (3.5), we have ⎧  ⎪ f j (; t) = tj , if 0 ≤ j ≤ 2 , ⎪ ⎪ ⎪ ⎪ ⎨ ft−1 (; t) = ft (; t) = 0 , " t ! t  ⎪ − f (; t) (x− 1)t− j j ⎪ j =3 j ⎪ " !  ⎪  ⎪ ⎩ = − tj =3 (−1) j tj − f j (; t) xt− j . Long f -vectors of face systems similar to those defined by (3.8) and (3.9) are discussed in Chapters 2 and 3 of Ref. [5]. Let us return to the symmetric cycle D, and associate with the decomposition set Q (T , D) for the vertex T the abstract simplicial complex  := (T , D) whose facet family is defined to be the family   E t − S(T , Q) : Q ∈ Q (T , D) , (3.10) where S(T , Q) is the separation set of the vertices T and Q. By construction, the complex  and the complex  defined by (3.7) coincide, and we come to the following result: Proposition 3.2. Let D be a symmetric cycle in a hypercube graph H (t, 2). Let T ∈ {1, −1}t be a vertex of the graph H (t, 2), such that for the unique inclusion-minimal subset of vertices Q (T , D) ⊂ V(D) with the property  Q=T , Q∈ Q (T , D)

we have q(T , D) ≥ 5 . (i) The components of the long f -vector f (; t) of the complex  : = (T , D) = (−T , D)

Dehn–Sommerville Type Relations for Vertex Decompositions

whose family of facets is defined by (3.10) satisfy the Dehn– Sommerville type relations ⎧ t ⎪ (; t) = f , if 0 ≤ j ≤ 2 , j ⎪ j ⎪ ⎪ ⎪ ⎨ ft−1 (; t) = ft (; t) = 0 , " t ! t  ⎪ − f j (; t) (x−1)t− j ⎪ j =3 j ⎪ !  " ⎪  ⎪ ⎩ = − t (−1) j t − f j (; t) xt− j . j =3

j

In particular, – if t := 5, then

t

f3 (; t) = – if t := 6, then f3 (; t) = f4 (; t) =

2

t 3t  2

−t =5;

− 2t + 4 = 12 , − 3t + 6 = 3 ;

– if t := 7, then   + 4t − 21 3t   = f5 (; t) + 4t − 2t + t   f4 (; t) = 2 f3 (; t) − 2 4t + 3t   = 2 f5 (; t) + 3t − 2 2t + 2t   f5 (; t) = f3 (; t) − 4t + 2t − t   = 12 f4 (; t) − 12 3t + 2t − t f3 (; t) =

1 f (; t) 2 4

=

1 2



 f4 (; t) + 35

= f5 (; t) + 21 , = 2 f3 (; t) − 35 = 2 f5 (; t) + 7 , = f3 (; t) − 21   = 12 f4 (; t) − 35 +14 ;

note that the quantity f4 (; 7) is always odd. (ii) For any j , 3 ≤ j ≤ t − 2, we have t j

− f j (; t) = −

j 

(−1)i

i =3

 t−i !t j −i

i

(iii) We have t−2  j =1

(−1) j f j (; t) = 0 .

" − fi (; t) .

45

46 Vertex Decompositions and Dehn–Sommerville Type Relations

Figure 3.2 A symmetric cycle D of the hypercube graph H (7, 2). For the vertex T := (−1, −1, 1, 1, 1, 1, 1) of the graph H (7, 2) we have T = Q0 + · · · + Q4 , for a unique inclusion-minimal subset Q (T , D) := {Q0 , . . . , Q4 } ⊂ V(D).

Example 3.3. Pick the vertex T := (−1, −1,

1,

1,

1,

1,

1)

of the hypercube graph H (7, 2). Consider a symmetric cycle D of the  graph H (7, 2), depicted in Figure 3.2. We have T = Q∈Q (T , D) Q, for a unique inclusion-minimal subset Q (T , D) := {Q0 , . . . , Q4 } ⊂ V(D), where Q0 : = ( 1, −1,

1,

1, −1,

1,

Q : = ( 1,

1,

1,

1,

1,

1, −1) ,

Q : = (−1,

1, −1,

1,

1,

1, −1) ,

1 2

Q : = (−1, −1, −1, −1,

1, −1,

1) ,

1, −1, −1, −1,

1) .

3

Q : = (−1, −1, 4

1) ,

The separation sets S(T , Qi ) are as follows: S(T , Q0 ) = {1, 5} , S(T , Q1 ) = {1, 2, 7} , S(T , Q2 ) = {2, 3, 7} , S(T , Q3 ) = {3, 4, 6} , S(T , Q4 ) = {4, 5, 6} . The facet family of the abstract simplicial complex  := (T , D) defined by (3.10) is %  {2, 3, 4, 6, 7} , {3, 4, 5, 6} , {1, 4, 5, 6} , {1, 2, 5, 7} , {1, 2, 3, 7}                E 7 −S(T , Q0 )

E 7 −S(T , Q1 )

E 7 −S(T , Q2 )

E 7 −S(T , Q3 )

E 7 −S(T , Q4 )

.

Dehn–Sommerville Type Relations for Vertex Decompositions

It is easy to check that the components of the long f -vector ⎞ ⎛ 3

1

5

↓

↓

7

↓

↓

f (; 7) = ⎝1, 7, 21, 22, 9, 1, 0, 0⎠ ↑

0

↑

2

↑

4

↑

6

of the complex  satisfy the relations given in Proposition 3.2.

47

Chapter 4

Vertex Decompositions in Hypercube Graphs, and Orthogonality Relations

If a vertex of a hypercube graph admits its large-size decomposition with respect to a symmetric cycle of the graph, then the face numbers of an abstract simplicial complex associated in a natural way with the decomposition set satisfy Dehn–Sommerville type relations; see the previous Chapter 3. In the present chapter, we consider large-size decomposition sets for vertices of hypercube graphs H (s, 2) and H (t, 2), where the parameters s and t have different parity. We show that in addition to the individual Dehn– Sommmerville type relations, enumerative descriptions of the decomposition sets for the vertices satisfy a certain common orthogonality relation. In Section 4.1 we recall Dehn–Sommerville type relations that are valid for large-size decomposition sets for vertices of a hypercube graph. In Section 4.2 we present a common orthogonality relation that establishes a connection between enumerative properties of large-size decomposition sets in the above graphs H (s, 2) and H (t, 2).

Symmetric Cycles Andrey O. Matveev c 2023 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4968-81-2 (Hardcover), 978-1-003-43832-8 (eBook) www.jennystanford.com

50 Vertex Decompositions and Orthogonality Relations

4.1 Dehn–Sommerville Type Relations for Vertex Decompositions in Hypercube Graphs. II Let H (s, 2) and H (t, 2) be the hypercube graphs on their vertex sets {1, −1}s and {1, −1}t , respectively. Throughout this chapter, we assume that s