Ranking patterns of unfolding models of codimension one Hidehiko - - PowerPoint PPT Presentation

Recent developments on geometric and algebraic methods in Economics August 24, 2014 Hokkaido University

Ranking patterns of unfolding models of codimension one

Hidehiko Kamiya (Nagoya University) Joint work with H. Terao and A. Takemura

This talk is based on

[1] H. Kamiya, A. Takemura, H. Terao, “Ranking patterns of unfolding models of codimension

• ne,” Advances in Applied Mathematics, Vol.

47, Iss. 2 (2011), pp. 379–400. [2] H. Kamiya, A. Takemura, H. Terao, “Arrangements stable under the Coxeter groups,” In: Configuration Spaces: Geometry, Combinatorics and Topology (ed. A. Bjorner,

• F. Cohen, C. De Concini, C. Procesi, M.

1

Salvetti), CRM Series, Vol. 14 (2012) pp. 327-354, Scuola Normale Superiore Pisa. 2

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

3

1 Introduction

• The unfolding model

(Coombs (1950), Psychol. Rev. 57, 145–158), also known as the ideal point model.

• A model for preference rankings.
• Objects 1, 2, . . . , m.
• An individual ranks these m objects.

4

Unfolding model

• The objects 1, 2, . . . , m are represented by

µ1, µ2, . . . , µm ∈ Rn.

• The individual is represented by

y ∈ Rn (his/her ideal point).

• Rn: the joint space.

5

Preference

• Preference: The nearer, the more preferred,

i.e., y prefers i to j ⇐ ⇒ ∥y − µi∥ < ∥y − µj∥.

• y has ranking ( i1
• best

i2

• 2nd best

· · · im

• worst object

) iff ∥y − µi1∥ < ∥y − µi2∥ < · · · < ∥y − µim∥. 6

Coffee Brands

Sourness Richness

Mocha Colombia Brazil Kilimanjaro Costa Rica Blue Mountain Hawaii Kona Mandheling Guatemala 1 4 2 4

★

y

y likes Mocha best, Blue Mountain next,.... 7

Admissible/inadmissible rankings

• In general, we can think of m! rankings

among m objects.

• Not all m! rankings are generated.

→ admissible/inadmissible rankings:            ∃y ∈ Rn s.t. ∥y − µi1∥ < · · · < ∥y − µim∥ → (i1i2 · · · im) : admissible; ∄y ∈ Rn s.t. ∥y − µi1∥ < · · · < ∥y − µim∥ → (i1i2 · · · im) : inadmissible. 8

Admissible Rankings

4 3 1 2

1 1 1 2 2 2 3 3 3 4 4 4

Inadmissible rankings: (1243) (1423) (2143) (2413) (4123) (4213) (4321) (3421) (4231) (2431) (2341) (2314) (2134) (1234) (3241) (3124) (1324) (3142) (1432) (4132) (4312) (3412) (1342) (3214)

2, m=4

18 out of 4! = 24 rankings are admissible. (For simplicity, µ1, µ2, µ3, µ4 are written as 1, 2, 3, 4.)

9

Ranking pattern

• Define the ranking pattern

RPUF(µ1, . . . , µm) := {admissible rankings}

• f the unfolding model with µ1, . . . , µm.
• The number of admissible rankings, i.e.,

|RPUF(µ1, . . . , µm)|, was obtained in Good and Tideman (1977), Kamiya and Takemura (1997), Zaslavsky (2002). 10

Different ranking patterns

• A different m-tuple (µ1, . . . , µm),

→ a different RPUF(µ1, . . . , µm).

4 3 1 2

1 1 1 2 2 2 3 3 3 4 4 4

Inadmissible rankings: (1243) (1423) (2143) (2413) (4123) (4213) (4321) (3421) (4231) (2431) (2341) (2314) (2134) (1234) (3241) (3124) (1324) (3142) (1432) (4132) (4312) (3412) (1342) (3214)

4 3 1 2

Inadmissible rankings: (2143) (2413) (2431) (4123) (4213) (4231)

3 3 3 2 2 2 4 4 4 1 1 1

(4321) (3421) (1423) (1243) (2341) (2314) (2134) (1234) (3241) (3124) (1324) (3142) (1432) (4132) (4312) (3412) (1342) (3214)

2, m=4

11

Problem

• How many ranking patterns are possible by

changing µ1, . . . , µm?

• When n ≥ m − 1, we have

RPUF(µ1, . . . , µm) = Pm (the set of all permutations of {1, . . . , m}).

• We are interested in the case 1 ≤ n ≤ m − 2.
• We are also interested in the case where we

ignore difference by a permutation of objects (“inequivalent” ranking patterns). 12

Problem of ranking patterns

• Thrall (1952), Michigan Math. J. 1, 81–88:

n = 1; upper bound; Young tableaux.

• Kamiya, Orlik, Takemura and Terao (2006),
• Ann. Comb. 10, 219–235:

n = 1; exact number; mid-hyperplane arrangement.

• This talk:

n = m − 2; exact number; inequivalent ranking patterns. 13

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

14

2 Unfolding as a braid slice

• Given µ1, . . . , µm ∈ Rn (not necessarily

n = m − 2).

• We will see:

RPUF(µ1, . . . , µm) can be obtained by slicing the “braid arrangement” Bm by an affine subspace K (a “braid slice”). 15

Affine subspace K

• Assume µ1, . . . , µm are normalized so that

∑m

i=1 µi = 0, (1/m) ∑m i=1 ∥µi∥2 = 1.

• Let

W :=     µT

1

. . . µT

m

    , u := − 1 2     ∥µ1∥2 − 1 . . . ∥µm∥2 − 1     ,

and define K := u + col W . 16

u colW K = u + colW V: x1+...+xm = 0

17

K ⊂ V

• Because of the normalization, we have

K ⊂ V := {(x1, . . . , xm)T ∈ Rm : x1 + · · · + xm = 0} ≃ Rm−1.

• All of our discussions will be in V .

18

Braid arrangement Bm in V

• Next, consider the braid arrangement Bm in

V : Bm := {Hij : 1 ≤ i < j ≤ m}, Hij := {(x1, . . . , xm)T ∈ V : xi = xj} ⊂ V. 19

x2 = x1 x2 = x3 x3 = x1 V: x1+x2+x3 =0

3

m=3:

20

Chambers of Bm

• Then, Bm cuts V into m! regions (called

chambers): Ci1···im = {(x1, . . . , xm)T ∈ V : xi1 > · · · > xim} ⊂ V, (i1 · · · im) ∈ Pm. 21

C123 C213 C132 C231 C312 C321 x2 = x1 x2 = x3 x3 = x1 x1>x2>x3 V: x1+x2+x3 =0

3

m=3:

22

Ci1···im intersecting K

• We can show: for each (i1 · · · im) ∈ Pm,

(i1 · · · im) is admissible ⇐ ⇒ K ∩ Ci1···im ̸= ∅.

• Let’s see a simple example.

23

Example: braid slice (m = 3, n = 1)

• m = 3, n = 1:

µ1= −3/ 14/3

1

√ µ2= 1/ 14/3 √ µ3= 2/ 14/3 √

• The ranking pattern is

1 2 2 3 1 3

(123) (213) (321) (231)

µ1 µ3 µ2

1

RPUF(µ1, µ2, µ3) = {(123), (213), (231), (321)}.

24

Example: braid slice (m = 3, n = 1)

• K in this case is

K =   −1 2   µ2

1 − 1

µ2

2 − 1

µ2

3 − 1

  + t   µ1 µ2 µ3   : t ∈ R    =    1 28   −13 11 2   + t   −3 1 2   : t ∈ R    ⊂ V. 25

C123 C213 C132 C231 C312 C321 x2 = x1 x2 = x3 x3 = x1 x1>x2>x3 V: x1+x2+x3 =0

3

m=3:

26

C123 C213 C132 C231 C312 C321 x2 = x1 x2 = x3 x3 = x1 K V: x1+x2+x3 =0

3

m=3:

27

C123 C213 C132 C231 C312 C321 V: x1+x2+x3 =0 x2 = x1 x2 = x3 x3 = x1

3

K

n=1, m=3:

1 2 2 3 1 3

(123) (213) (321) (231)

µ1 µ3 µ2

1

28

Proof

• The proof is quite simple:

∥y − µi1∥ < · · · < ∥y − µim∥, ∃y ∈ Rn ⇐ ⇒ µT

i1y − 1

2(∥µi1∥2 − 1) > · · · > µT

imy − 1

2(∥µim∥2 − 1), ∃y ∈ Rn ⇐ ⇒ W y + u ∈ Ci1···im, ∃y ∈ Rn ⇐ ⇒ K ∩ Ci1···im ̸= ∅. 29

Unfolding model as a braid slice by K

• Therefore,

Proposition: Unfolding as a braid slice ✓ ✏ Ranking patterns of unfolding models can be obtained as RPUF(µ1, . . . , µm) = {(i1 · · · im) : K ∩ Ci1···im ̸= ∅}. ✒ ✑ 30

Unfolding model as a braid slice by K

• In this way, RPUF(µ1, . . . , µm) can be
• btained by “slicing” Bm by K.

C123 C213 C132 C231 C312 C321 V: x1+x2+x3 =0 x2 = x1 x2 = x3 x3 = x1

3

K

n=1, m=3:

(123) (213) (321) (231)

µ1 µ3 µ2

1

31

Codimension one: n = m − 2

• Hereafter, we will assume n = m − 2, and

that µ1, . . . , µm are “generic.”

• Then, dim K = n = m − 2 = dim V − 1.
• Hence, in this case, we will say the unfolding

model is of codimension one.

• We can also see 0 /

∈ K.

• Thus,

K is an affine hyperplane in V . 32

Arbitrary affine hyperplane K˜

v

• Let’s forget the unfolding model for a

moment.

• In general, an arbitrary affine hyperplane in V

can be indexed by its normal vector ˜ v ∈ V : K˜

v := {x ∈ V : ˜

vT (x − ˜ v) = 0}, ˜ v ̸= 0. 33

V

v ~ Kv

~

34

Braid slice RP(˜ v) by K˜

v

• Let’s consider slicing Bm by K˜

v:

RP(˜ v) := {(i1 · · · im) : K˜

v ∩ Ci1···im ̸= ∅}.

• We call RP(˜

v) (the ranking pattern of) the braid slice by K˜

v.

35

C123 C213 C132 C231 C312 C321 Kv

~

v ~ RP(v) = {(213), (231), (321), (312)} ~

36

K = Kv of unfolding model

• Getting back to the unfolding model, we can

write K = u + col W as K = Kv with v := u − projcol W (u) = { Im − W (W T W )−1W T } u. 37

v u colW K = u + colW = Kv V: x1+...+xm = 0

38

RPUF(µ1, . . . , µm) = RP(v(µ1, . . . , µm))

• v is a function of µ1, . . . , µm:

v = v(µ1, . . . , µm).

• We can write RPUF(µ1, . . . , µm) as

RPUF(µ1, . . . , µm) = {(i1 · · · im) : Kv(µ1,...,µm) ∩ Ci1···im ̸= ∅} = RP(v(µ1, . . . , µm)). 39

Stability of RP(v(µ1, . . . , µm))

• m-tuple (µ1, . . . , µm) changes just a little,

→ v = v(µ1, . . . , µm) and thus Kv change just a little, → the braid slice by Kv remains exactly the same.

• (µ1, . . . , µm) changes beyond a certain

extent, → the braid slice changes. 40

v D Kv

C213 C231 C312 C321

v’ D’ Kv’

C123 C213 C231 C321

41

Stability of RP(˜ v), ˜ v ̸= 0

• Let’s investigate the stability of the braid

slices RP(˜ v), ˜ v ̸= 0, in general. 42

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

43

3 All braid slices

• We will investigate the stability of the braid

slices RP(˜ v), ˜ v ̸= 0, and find all (different) braid slices. 44

All-subset arrangement Am

• First, define the all-subset arrangement Am

in V by Am := {HI : I ⊂ {1, . . . , m}, 1 ≤ |I| ≤ m − 1}, HI := {(x1, . . . , xm)T ∈ V : ∑

i∈I

xi = 0} ⊂ V. 45

x2 = 0 ( x1+x3 = 0) H{2} = H{1,3} x1 = 0 ( x2+x3 = 0) H{1} = H{2,3} x3 = 0 ( x1+x2 = 0) H{3} = H{1,2} V: x1+x2+x3 =0

m=3:

3

46

Chambers of Am

• Let Ch(Am) := {chambers D of Am}.

D

47

Generic braid slices

• We only consider RP(˜

v) for “regular” ˜ v: ˜ v ∈ V \ ∪

H∈Am

H = ⊔

D∈Ch(Am)

D.

• We will say the braid slices are “generic” in

this case. 48

RPD := RP(˜ v), ∀˜ v ∈ D

• For ˜

v’s in a chamber D ∈ Ch(Am), RP(˜ v) are the same: RP(˜ v) : the same for all ˜ v ∈ D. We can define RPD := RP(˜ v), ˜ v ∈ D.

• For different chambers D and D′, RP(˜

v) changes: RPD ̸= RPD′ for D ̸= D′. 49

~

v D

~

Kv

C213 C231 C312 C321

v’

~

D’

~

Kv’

C123 C213 C231 C321

50

{chambers of Am} ↔ {generic braid slices}

• Therefore,

Proposition: {D} ↔ {RPD} ✓ ✏ We have a bijection Ch(Am) → {RPD : D ∈ Ch(Am)}, D → RPD = RP(˜ v), ˜ v ∈ D. ✒ ✑ 51

v(µ1, . . . , µm) ∈ V \ ∪

H∈Am H

• Let’s get back to the unfolding model.
• We can show

v(µ1, . . . , µm) ∈ V \ ∪

H∈Am

H = ⊔

D∈Ch(Am)

D.

D v(µ1,..., µm)

52

RP(v(µ1, . . . , µm)): generic

• Thus,

RPUF(µ1, . . . , µm) = RP(v(µ1, . . . , µm)) is a generic braid slice: RPUF(µ1, . . . , µm) = RPD for D ∋ v(µ1, . . . , µm). 53

D

v(µ1,..., µm)

54

D not attained by v( · , . . . , · )

• However, not all D ∈ Ch(Am) can be

attained by v( · , . . . , · ).

• Hence,

{r.p.’s of unfoldings} {generic braid slices}.

• Not all generic braid slices can be realized by

the unfolding model. 55

C123 C213 C132 C231 C312 C321 Kv

~

v ~ RP(v) = {(213), (231), (321), (312)} ~

56

D not attained by v( · , . . . , · )

• We have to identify Ds which can/cannot be

attained by v( · , . . . , · ). 57

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

58

4 Unrealizable braid slices

• We will identify D ∈ Ch(Am) which cannot

be attained by v( · , . . . , · ), and thereby identify unrealizable braid slices RPD. 59

Image of v( · , . . . , · )

• We can show im v := {v(µ1, . . . , µm) :

generic µ1, . . . , µm ∈ Rm−2} is given by im v = ⊔

D∈Ch(Am), D̸=−D1,...,−Dm

D where −D1, . . . , −Dm ∈ Ch(Am) are −Di := {v ∈ V : vi < 0, vj > 0 for all j ̸= i}. 60

−D1 −D3 −D2 (x1 < 0, x2,x3 > 0) (x2 < 0, x1,x3 > 0) (x3 < 0, x1,x2 > 0)

x2 = 0 ( x1+x3 = 0) H{2} = H{1,3} x1 = 0 ( x2+x3 = 0) H{1} = H{2,3} x3 = 0 ( x1+x2 = 0) H{3} = H{1,2} V: x1+x2+x3 =0

m=3:

3

61

Unrealizable braid slices

• Therefore,

Theorem: Unrealizable braid slices ✓ ✏ RPD cannot be realized by the unfolding model if and only if D = −Di for some i = 1, . . . , m. ✒ ✑ 62

Unrealizable braid slices

• That is, exactly

RP−D1, . . . , RP−Dm are unrealizable braid slices. 63

−D1 −D3 −D2 (x1 < 0, x2,x3 > 0) (x2 < 0, x1,x3 > 0) (x3 < 0, x1,x2 > 0)

x2 = 0 ( x1+x3 = 0) H{2} = H{1,3} x1 = 0 ( x2+x3 = 0) H{1} = H{2,3} x3 = 0 ( x1+x2 = 0) H{3} = H{1,2} V: x1+x2+x3 =0

m=3:

3

64

C123 C213 C132 C231 C312 C321 Kv

~

v ~ RP(v) = {(213), (231), (321), (312)} ~

65

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

66

5 Number of ranking patterns

• So far, we have seen

{r.p.’s of unfoldings} {generic braid slices} ↕ {chambers of Am} and {generic braid slices} \ {r.p.’s of unfoldings} = {RP−D1, . . . , RP−Dm}. 67

Main result 1 Theorem: Number of r.p.’s ✓ ✏ The number q(m) of ranking patterns of unfolding models of codimension one with m objects is q(m) = |{chambers of Am}| − m. ✒ ✑ 68

q(m), m ≤ 8 Corollary: q(m) for m ≤ 8 ✓ ✏ For m ≤ 8, we have q(3) = 3, q(4) = 28, q(5) = 365, q(6) = 11286, q(7) = 1066037, q(8) = 347326344. ✒ ✑ 69

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

70

6 Inequivalent ranking patterns

• Let’s consider the case where we ignore

difference by a permutation of objects.

• We consider two ranking patterns are

essentially the same if they are just the relabelings of the objects of each other.

• What is the number of essentially different

ranking patterns of unfolding models of codimension one? 71

Equivalence of ranking patterns

• Let Sm be the symmetric group on m letters

(all bijections σ : {1, . . . , m} → {1, . . . , m}).

• We will say RPD and RPD′ are equivalent:

RPD ∼ RPD′ if and only if RPD = σRPD′, ∃σ ∈ Sm, where σRPD′ := {(σ(i1) · · · σ(im)) : (i1 · · · im) ∈ RPD′}. 72

Action of Sm on Ch(Am)

• Consider the action of Sm on Ch(Am):

Sm × Ch(Am) ∋ (σ, D) − → σD ∈ Ch(Am), where σD := {σv : v ∈ D} with σv := (vσ−1(1), . . . , vσ−1(m))T for v = (v1, . . . , vm)T . 73

RPD ∼ RPD′ ⇐ ⇒ O(D) = O(D′)

• We can check

RPσD = σRPD.

• Thus,

RPD ∼ RPD′ ⇐ ⇒ D = σD′, ∃σ ∈ Sm (i.e., O(D) = O(D′)), where O(D) := {σD : σ ∈ Sm} is the Sm-orbit containing D. 74

RPD ∼ RPD′ ⇐ ⇒ O(D) = O(D′)

• Therefore, the number of inequivalent braid

slices RPD, D ∈ Ch(Am), is equal to |{Sm-orbits of Ch(Am)}|. 75

Orbit of excluded chambers

• On the other hand, the excluded m chambers

−D1, . . . , −Dm constitute one orbit O(−Dm) = {−D1, . . . , −Dm}. 76

Number of inequivalent ranking patterns

• Therefore,

Proposition: # of inequivalent r.p.’s ✓ ✏ The number of inequivalent ranking pat- terns of unfolding models of codimension

• ne is

|{Sm-orbits of Ch(Am)}| − 1. ✒ ✑ 77

|{Sm-orbits of Ch(Am)}|

• Our job is to count

|{Sm-orbits of Ch(Am)}|. 78

{F ∈ Ch(Am ∪ Bm) : F ⊂ C1···m}

• There is a one-to-one correspondence

{F ∈ Ch(Am ∪ Bm) : F ⊂ C1···m} ← → {Sm-orbits of Ch(Am)} given by F − → O(DF ), where DF := unique D ∈ Ch(Am) such that F ⊂ D for F ∈ Ch(Am ∪ Bm). 79

F ↔ O(DF ), m = 3

x2 = x1 x2 = x3 x3= x1 x2 =0 (x1+x3 = 0) x1 =0 (x2+x3 = 0) x3 =0 (x1+x2 = 0)

3 3

m =3:

F F ’

F

F C123

x2 = x1 x2 = x3 x3 = x1 x2 =0 (x1+x3 = 0) x1 =0 (x2+x3 = 0) x3 =0 (x1+x2 = 0)

3 3

x3 =0 (

m =3:

DF (DF) | (DF)|=3

{Sm-orbits of Ch(Am)} = {O(DF ), O(DF ′)}. 80

|{Sm-orbits of Ch(Am)}|

• Since |Ch(Bm)| = |Sm| = m!, we obtain

|{Sm-orbits of Ch(Am)}| = |{F ∈ Ch(Am ∪ Bm) : F ⊂ C1···m}| = |Ch(Am ∪ Bm)| m! . 81

Main result 2 Theorem: # of inequivalent r.p.’s ✓ ✏ The number qIE(m) of inequivalent rank- ing patterns of unfolding models of codi- mension one is qIE(m) = |Ch(Am ∪ Bm)| m! − 1. ✒ ✑ 82

qIE(m), m ≤ 9 Corollary: qIE(m) for m ≤ 9 ✓ ✏ For m ≤ 9, we have qIE(3) = 1, qIE(4) = 3, qIE(5) = 11, qIE(6) = 55, qIE(7) = 575, qIE(8) = 16639, qIE(9) = 1681099. ✒ ✑ 83

Illustration: m = 4

• We illustrate our results when m = 4.

84

F, F ′, ˜ F, ˜ F ′ ⊂ C1234

• Chamber

C1234 : x1 > x2 > x3 > x4

• f B4 contains exactly 4 chambers

F, F ′, ˜ F, ˜ F ′ ∈ Ch(A4 ∪ B4). 85

F, F ′, ˜ F, ˜ F ′ ⊂ C1234

C1234: x1 > x2 > x3 > x4

x2>x4 x1+x4>0 x1>x2 x1>x4 x1>x3 x3>x4 x2>x3 x1+x3>0 x1+x2>0 x1>0 x4>0 x2>0 x3>0 F ’ F ~ F ’ F ~ x2>x4 x1>x2 x1>x4 x1>x3 x3>x4 x2>x3

C1234

F ~ ’ F F ’ F ~ , , , ⊂ C1234

∪

4 4 4

(intersection with the unit sphere S2 in V ≃ R3)

86

x2>x4 x1+x4>0 x1>x2 x1>x4 x1>x3 x3>x4 x2>x3 x1+x3>0 x1+x2>0 x1>0 x4>0 x2>0 x3>0 F ’ F ~ F ’ F ~

A4 ∪ B4 (intersection with the unit sphere S2 in V ≃ R3)

87

DF , DF ′, D ˜

F , D ˜ F ′ ∈ Ch(A4)

• The chambers of A4 containing F, F ′, ˜

F, ˜ F ′ are DF = −D4 : x1 > 0, x2 > 0, x3 > 0, DF ′ = D1 : x4 < 0, x3 < 0, x2 < 0, D ˜

F = E : x2 + x3 > 0, x1 + x3 > 0, x3 < 0,

D ˜

F ′ = E′ : x3 + x2 < 0, x4 + x2 < 0, x2 > 0.

88

DF , DF ′, D ˜

F , D ˜ F ′ ∈ Ch(A4)

x1+x3>0 x3<0 x2+x3>0 x4+x2<0 x3>0 x1>0 x2>0 F

~ D =E

F

~

F D = －D4

F

x

4

< x

2

< x

3

<

F ’

F’

D = D1 D =E’

F ’

~

F ’

~

x2>0 x3+x2<0

89

Cross section

• These DF , DF ′, D ˜

F , D ˜ F ′ constitute a

complete set of representatives of the orbits:

{Sm-orbits of Ch(Am)} = {O(DF ), O(DF ′), O(D ˜

F ), O(D ˜ F ′)}.

90

Four inequivalent braid slices

• Four inequivalent braid slices:

RPDF = RP−D4, RPDF ′ = RPD1, RPD ˜

F = RPE, RPD ˜ F ′ = RPE′.

– RP−D4: not realizable. – RPD1, RPE, RPE′: realizable. 91

Three inequivalent r.p.’s of unfoldings

• There are exactly

three inequivalent ranking patterns RPD1, RPE, RPE′

• f unfolding models.

92

Unrealizable RP−D4

• Unrealizable braid slice:

RP−D4 = P4 \ {(4123), (4132), (4213), (4231), (4312), (4321)}. (“object 4 is never ranked first”)

93

Realizable RPD1, RPE, RPE′

• Realizable braid slices:

RPD1 = P4 \ {(2341), (2431), (3241), (3421), (4231), (4321)}, (“object 1 is never ranked last”) RPE = P4 \ {(3412), (3421), (4312), (4321), (4132), (4231)}, RPE′ = P4 \ {(3412), (4312), (3421), (4321), (3241), (4231)}.

94

95

Realizing unfolding models

• The unfolding models realizing

RPD1, RPE, RPE′ are illustrated in the following figures (µ1, µ2, µ3, µ4 are written as 1, 2, 3, 4 in the figures). 96

RPD1

4 1 2 3

2 2 2 3 3 3 1 1 1 4 4 4

Inadmissible rankings: (2341) (2431) (3241) (3421) (4231) (4321) (4132) (1432) (4312) (3412) (3142) (3124) (3214) (2314) (1342) (1234) (2134) (1243) (2413) (4213) (4123) (1423) (2143) (1324)

97

RPE

2 2 2 4 4 4 3 3 3 1 1 1

3 2 1 4

(1423) (1432) (1342) (3124) (3142) (3214) (3241) (2341) (2431) (2413) (4213) (4123) (1243) (1324) (1234) (2314) (2134) (2143) Inadmissible rankings: (3412) (3421) (4312) (4321) (4132) (4231)

98

RPE′

4 1 2 3

Inadmissible rankings: (3412) (4312) (3421) (4321) (3241) (4231)

1 1 1 3 3 3 4 4 4 2 2 2

(4132) (1432) (2431) (2341) (3142) (3124) (3214) (2314) (1342) (1234) (2134) (1243) (2413) (4213) (4123) (1423) (2143) (1324)

99

Table of Contents

• 1. Introduction
• 2. Unfolding as a braid slice
• 3. All braid slices
• 4. Unrealizable braid slices
• 5. Number of ranking patterns
• 6. Inequivalent ranking patterns
• 7. Concluding remarks

100

7 Concluding remarks

• We found the number of ranking patterns of

unfolding models when n = m − 2.

• Open problem: What is the number of

ranking patterns when 1 < n < m − 2? 101

Acknowledgment: H. Kamiya was partially supported by JSPS KAKENHI Grant Numbers 22540134, 25400201. 102

Thank you.

103

Notes

• ˜

F ′ is obtained from ˜ F by changing (x1, . . . , x4) → (−x4, . . . , −x1).

• RPE′ can be obtained from RPE by

reversing the indices : i → 5 − i (i = 1, . . . , 4) and taking the opposite rankings : (i1i2i3i4) → (i4i3i2 104

• RPE and RPE′ are not equivalent.
• The same is true for F and F ′ (E and E′).

105

• Orbit sizes are

(|O(−D4)| = 4) |O(D1)| = 4, |O(E)| = |O(E′)| = (4 3 ) × 3 = 12, so q(4) = 28 (mentioned ealier) can also be confirmed by q(4) = 4 + 2 × 12 = 28. 106

• Good and Tideman (1977), J. Combin.

Theory A 23, 34–45;

• Kamiya and Takemura (1997), J. Multivariate
• Anal. 61,1–28;
• Zaslavsky (2002), Discrete Comput. Geom.

27, 303–351. 107