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![Page 1: Arrangements and Combinatorics - MITrstan/transparencies/comb.pdf · The main reference An introduction to hyperplane arrangements, in Geometric Combinatorics (E. Miller, V. Reiner,](https://reader035.vdocument.in/reader035/viewer/2022071102/5fdbab06a02f1b604e228da5/html5/thumbnails/1.jpg)
Arrangements and Combinatorics
Richard P. Stanley
M.I.T.
Arrangements and Combinatorics – p. 1
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The main reference
An introduction to hyperplane arrangements,in Geometric Combinatorics (E. Miller, V. Reiner,and B. Sturmfels, eds.), IAS/Park CityMathematics Series, vol. 13, AmericanMathematical Society, Providence, RI, 2007,pp. 389–496.
math.mit.edu/∼rstan/arrangements/arr.html
Arrangements and Combinatorics – p. 2
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Posets
A poset (partially ordered set) is a set P andrelation ≤ satisfying ∀x, y, z ∈ P :
(P1) (reflexivity) x ≤ x
(P2) (antisymmetry) If x ≤ y and y ≤ x, thenx = y.
(P3) (transitivity) If x ≤ y and y ≤ z, then x ≤ z.
Arrangements and Combinatorics – p. 3
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Arrangements
K : a field
A : a (finite) arrangement in V = Kn
rk(A) (rank of A) : dimension of space
spanned by normals to H ∈ A
Arrangements and Combinatorics – p. 4
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Subspaces X, Y, W
Y = any complement to subspace X of Kn
spanned by normals to H ∈ A
W = {v ∈ V : v · y = 0 ∀y ∈ Y }.
If char(K) = 0 can take W = X.
Arrangements and Combinatorics – p. 5
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Essentialization
codimW (H ∩ W ) = 1, ∀H ∈ A
Essentialization of A:
ess(A) = {H ∩ W : H ∈ A},
an arrangment in W .
rk(ess(A)) = rk(A)
A is essential if ess(A) = A, i.e.,rk(A) = dim(A).
Arrangements and Combinatorics – p. 6
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Essentialization
codimW (H ∩ W ) = 1, ∀H ∈ A
Essentialization of A:
ess(A) = {H ∩ W : H ∈ A},
an arrangment in W .
rk(ess(A)) = rk(A)
A is essential if ess(A) = A, i.e.,rk(A) = dim(A).
Arrangements and Combinatorics – p. 6
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Essentialization
codimW (H ∩ W ) = 1, ∀H ∈ A
Essentialization of A:
ess(A) = {H ∩ W : H ∈ A},
an arrangment in W .
rk(ess(A)) = rk(A)
A is essential if ess(A) = A, i.e.,rk(A) = dim(A).
Arrangements and Combinatorics – p. 6
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Example of essentialization
A ess( )A
W
Arrangements and Combinatorics – p. 7
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The intersection poset
L(A) : nonempty intersections of hyperplanesin A, ordered by reverse inclusion
Include V as the bottom element of L(A),denoted 0̂.
Note. L(A) ∼= L(ess(A))
L(A) is the most important combinatorial objectassociated with A.
Arrangements and Combinatorics – p. 8
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The intersection poset
L(A) : nonempty intersections of hyperplanesin A, ordered by reverse inclusion
Include V as the bottom element of L(A),denoted 0̂.
Note. L(A) ∼= L(ess(A))
L(A) is the most important combinatorial objectassociated with A.
Arrangements and Combinatorics – p. 8
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Examples of intersection posets
Arrangements and Combinatorics – p. 9
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Rank function
Chain of length k: x0 < x1 < · · · < xk
Graded poset of rank n: every maximal chainhas length n
Rank function: ρ(x) is the length k of longestchain x0 < x1 < · · · < xk = x.
Arrangements and Combinatorics – p. 10
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Rank function on L(A)
Proposition. L(A) is graded of rank equal tork(A). Rank function:
rk(x) = codim(x) = n − dim(x),
where dim(x) is the dimension of x as an affinesubspace of V .
Proof. Straightforward. �
Arrangements and Combinatorics – p. 11
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Rank function on L(A)
Proposition. L(A) is graded of rank equal tork(A). Rank function:
rk(x) = codim(x) = n − dim(x),
where dim(x) is the dimension of x as an affinesubspace of V .
Proof. Straightforward. �
Arrangements and Combinatorics – p. 11
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Example of L(A)
ap
V = R
bca
p q
1
2
1
0
0
2
rank dim
2
b
cq
Arrangements and Combinatorics – p. 12
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The Möbius function
P is locally finite: every interval[x, y] = {z : x ≤ z ≤ y} is finite.
Int(P ) = set of (nonempty) intervals of P
Define µ = µP : Int(P ) → Z (the Möbiusfunction of P ) by:
µ(x, x) = 1, for all x ∈ P
µ(x, y) = −∑
x≤z<y
µ(x, z), for all x < y in P.
Write µ(x) = µ(0̂, x).
Arrangements and Combinatorics – p. 13
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The Möbius function
P is locally finite: every interval[x, y] = {z : x ≤ z ≤ y} is finite.
Int(P ) = set of (nonempty) intervals of P
Define µ = µP : Int(P ) → Z (the Möbiusfunction of P ) by:
µ(x, x) = 1, for all x ∈ P
µ(x, y) = −∑
x≤z<y
µ(x, z), for all x < y in P.
Write µ(x) = µ(0̂, x).
Arrangements and Combinatorics – p. 13
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The Möbius function
P is locally finite: every interval[x, y] = {z : x ≤ z ≤ y} is finite.
Int(P ) = set of (nonempty) intervals of P
Define µ = µP : Int(P ) → Z (the Möbiusfunction of P ) by:
µ(x, x) = 1, for all x ∈ P
µ(x, y) = −∑
x≤z<y
µ(x, z), for all x < y in P.
Write µ(x) = µ(0̂, x).Arrangements and Combinatorics – p. 13
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Example of Möbius function
1
−1 −1 −1
112
−2
−1
1
Numbers denote µ(x).
Arrangements and Combinatorics – p. 14
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Möbius inversion formula
P = finite poset
f, g : P → L (a field, or even just an abeliangroup)
Theorem. Equivalent:
f(x) =∑
y≥x
g(y), for all x ∈ P
g(x) =∑
y≥x
µ(x, y)f(y), for all x ∈ P.
Proof. Straightforward. �
Arrangements and Combinatorics – p. 15
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Möbius inversion formula
P = finite poset
f, g : P → L (a field, or even just an abeliangroup)
Theorem. Equivalent:
f(x) =∑
y≥x
g(y), for all x ∈ P
g(x) =∑
y≥x
µ(x, y)f(y), for all x ∈ P.
Proof. Straightforward. �Arrangements and Combinatorics – p. 15
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The characteristic polynomial
Definition. The characteristic polynomial χA(t)of the arrangement A is defined by
χA(t) =∑
x∈L(A)
µ(x)tdim(x).
Note. x = V contributes tn, and each H ∈ Acontributes −tn−1. Hence
χA(t) = tn − (#A)tn−1 + · · · .
Arrangements and Combinatorics – p. 16
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The characteristic polynomial
Definition. The characteristic polynomial χA(t)of the arrangement A is defined by
χA(t) =∑
x∈L(A)
µ(x)tdim(x).
Note. x = V contributes tn, and each H ∈ Acontributes −tn−1. Hence
χA(t) = tn − (#A)tn−1 + · · · .
Arrangements and Combinatorics – p. 16
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An example
Example.
1
−1
112
−2
1
−1−1−1
χA(t) = t3 − 4t2 + 5t − 2 = (t − 1)2(t − 2).
Arrangements and Combinatorics – p. 17
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The boolean algebra
Suppose all hyperplanes in A are linearlyindependent, and #A = n. Then all intersectionsare nonempty and distinct, so
L(A) ∼= Bn,
the boolean algebra of all subsets of[n] = {1, . . . , n}, ordered by inclusion.
Arrangements and Combinatorics – p. 18
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Characteristic polynomial of Bn
Easy induction argument: µ(0̂, x) = (−1)n−dim x.Hence
χA(t) =n∑
i=0
(n
i
)(−1)n−iti = (t − 1)n.
Arrangements and Combinatorics – p. 19
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Regions
Let K = R. Region (or chamber) of A:
connected component of Rn −⋃
H∈A H.
r(A) = number of regions of A
A region R of A is relatively bounded if it
becomes bounded in ess(A).
b(A) = number of relatively bounded regions of A
Arrangements and Combinatorics – p. 20
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Example of r(A) and b(A)
r(A) = 10, b(A) = 2
Arrangements and Combinatorics – p. 21
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Zaslavsky’s theorem (1975)
Current goal:
Theorem. Let A be an arrangement of rank r inRn. Then
r(A) = (−1)nχA(−1)
b(A) = (−1)rχA(1).
Proof will be by induction on #A (the number ofhyperplanes).
Arrangements and Combinatorics – p. 22
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Zaslavsky’s theorem (1975)
Current goal:
Theorem. Let A be an arrangement of rank r inRn. Then
r(A) = (−1)nχA(−1)
b(A) = (−1)rχA(1).
Proof will be by induction on #A (the number ofhyperplanes).
Arrangements and Combinatorics – p. 22
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Subarrangements and restrictions
subarrangement of A: a subset B ⊆ A
For x ∈ L(A) define
Ax = {H ∈ A : x ⊆ H} ⊆ A
Also define the restriction of A to x to be thearrangement in the affine space A:
Ax = {x ∩ H 6= ∅ : H ∈ A−Ax}.
Arrangements and Combinatorics – p. 23
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Subarrangements and restrictions
subarrangement of A: a subset B ⊆ A
For x ∈ L(A) define
Ax = {H ∈ A : x ⊆ H} ⊆ A
Also define the restriction of A to x to be thearrangement in the affine space A:
Ax = {x ∩ H 6= ∅ : H ∈ A−Ax}.
Arrangements and Combinatorics – p. 23
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L(Ax) and L(Ax)
Note that if x ∈ L(A), then
L(Ax) ∼= Λx := {y ∈ L(A) : y ≤ x}
L(Ax) ∼= Vx := {y ∈ L(A) : y ≥ x}.
Arrangements and Combinatorics – p. 24
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Example of Ax and Ax
AKA
x
xA
K
K
Arrangements and Combinatorics – p. 25
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Triple of arrangments
Choose H0 ∈ A. Define
A′ = A− {H0}
A′′ = AH0.
Call (A, A′, A′′) a triple of arrangements withdistinguished hyperplane H0.
Arrangements and Combinatorics – p. 26
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Triple of arrangments
Choose H0 ∈ A. Define
A′ = A− {H0}
A′′ = AH0.
Call (A, A′, A′′) a triple of arrangements withdistinguished hyperplane H0.
Arrangements and Combinatorics – p. 26
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Recurrence for r(A) and b(A)
Lemma. Let (A,A′,A′′) be a triple of realarrangements with distinguished hyperplane H0.Then
r(A) = r(A′) + r(A′′)
b(A) =
{b(A′) + b(A′′), if rk(A) = rk(A′)
0, if rk(A) = rk(A′) + 1.
Arrangements and Combinatorics – p. 27
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The case rk(A) = rk(A′) + 1
0H
Arrangements and Combinatorics – p. 28
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Proof of lemma (sketch)
Note that r(A) equals r(A′) plus the number ofregions of A′ cut into two regions by H0. Easy togive a bijection between regions of A′ cut in twoby H0 and regions of A′′, proving
r(A) = r(A′) + r(A′′).
Proof of recurrence for b(A) analogous. �
Arrangements and Combinatorics – p. 29
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Proof of lemma (sketch)
Note that r(A) equals r(A′) plus the number ofregions of A′ cut into two regions by H0. Easy togive a bijection between regions of A′ cut in twoby H0 and regions of A′′, proving
r(A) = r(A′) + r(A′′).
Proof of recurrence for b(A) analogous. �
Arrangements and Combinatorics – p. 29
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The deletion-restriction recurrence
Lemma. Let (A,A′,A′′) be a triple of realarrangements. Then
χA(t) = χA′(t) − χA′′(t).
Zaslavsky’s theorem (r(A) = (−1)nχA(−1)) is animmediate consequence of above lemma andthe recurrence r(A) = r(A′) + r(A′′).
The proof for b(A) is analogous but a little morecomplicated.
Arrangements and Combinatorics – p. 30
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The deletion-restriction recurrence
Lemma. Let (A,A′,A′′) be a triple of realarrangements. Then
χA(t) = χA′(t) − χA′′(t).
Zaslavsky’s theorem (r(A) = (−1)nχA(−1)) is animmediate consequence of above lemma andthe recurrence r(A) = r(A′) + r(A′′).
The proof for b(A) is analogous but a little morecomplicated.
Arrangements and Combinatorics – p. 30
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The deletion-restriction recurrence
Lemma. Let (A,A′,A′′) be a triple of realarrangements. Then
χA(t) = χA′(t) − χA′′(t).
Zaslavsky’s theorem (r(A) = (−1)nχA(−1)) is animmediate consequence of above lemma andthe recurrence r(A) = r(A′) + r(A′′).
The proof for b(A) is analogous but a little morecomplicated.
Arrangements and Combinatorics – p. 30
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Whitney’s theorem
To prove: χA(t) = χA′(t) − χA′′(t).
Basic tool (H. Whitney, 1935, for lineararrangements). A subarrangement B ⊆ A iscentral if
⋂H∈B H 6= ∅.
Theorem. Let A be an arrangement in ann-dimensional vector space. Then
χA(t) =∑
B⊆AB central
(−1)#Btn−rk(B).
Arrangements and Combinatorics – p. 31
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Whitney’s theorem
To prove: χA(t) = χA′(t) − χA′′(t).
Basic tool (H. Whitney, 1935, for lineararrangements). A subarrangement B ⊆ A iscentral if
⋂H∈B H 6= ∅.
Theorem. Let A be an arrangement in ann-dimensional vector space. Then
χA(t) =∑
B⊆AB central
(−1)#Btn−rk(B).
Arrangements and Combinatorics – p. 31
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Example of Whitney’s theorem
c d
a
b
B #B rk(B)
∅ 0 0a 1 1b 1 1c 1 1d 1 1
ac 2 2ad 2 2
bc 2 2bd 2 2cd 2 2
acd 3 2
⇒ χA(t) = t2 − 4t + (5 − 1) = t2 − 4t + 4.
Arrangements and Combinatorics – p. 32
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The crosscut theorem
Easy fact: Every interval [0̂, z] of L(A) is alattice, i.e., any two elements x, y have a meet(greatest lower bound) x ∧ y and join (leastupper bound) x ∨ y.
Lemma (crosscut theorem for L(A)). For allz ∈ L(A),
µ(z) =∑
B⊆Az
z=⋂
H∈BH
(−1)#B.
Arrangements and Combinatorics – p. 33
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The crosscut theorem
Easy fact: Every interval [0̂, z] of L(A) is alattice, i.e., any two elements x, y have a meet(greatest lower bound) x ∧ y and join (leastupper bound) x ∨ y.
Lemma (crosscut theorem for L(A)). For allz ∈ L(A),
µ(z) =∑
B⊆Az
z=⋂
H∈BH
(−1)#B.
Arrangements and Combinatorics – p. 33
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Proof of Whitney’s theorem
Lemma (crosscut theorem for L(A)). For allz ∈ L(A),
µ(z) =∑
B⊆Az
z=⋂
H∈BH
(−1)#B.
Note that z =⋂
H∈B H implies thatrk(B) = n − dim z. Multiply both sides by tdim(z)
and sum over z to obtain
χA(t) =∑
B⊆AB central
(−1)#Btn−rk(B). �
This proves Zaslavsky’s theorem.
Arrangements and Combinatorics – p. 34
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Proof of Whitney’s theorem
Lemma (crosscut theorem for L(A)). For allz ∈ L(A),
µ(z) =∑
B⊆Az
z=⋂
H∈BH
(−1)#B.
Note that z =⋂
H∈B H implies thatrk(B) = n − dim z. Multiply both sides by tdim(z)
and sum over z to obtain
χA(t) =∑
B⊆AB central
(−1)#Btn−rk(B). �
This proves Zaslavsky’s theorem.
Arrangements and Combinatorics – p. 34
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Alternative formulation
Later: coefficients of χA(t) alternate in sign.More strongly, if rk(x) = i then
(−1)iµ(x) > 0.
Thus:
r(A) =∑
x∈LA
|µ(x)|
b(A) =
∣∣∣∣∣∑
x∈LA
µ(x)
∣∣∣∣∣ .
Arrangements and Combinatorics – p. 35
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Alternative formulation
Later: coefficients of χA(t) alternate in sign.More strongly, if rk(x) = i then
(−1)iµ(x) > 0.
Thus:
r(A) =∑
x∈LA
|µ(x)|
b(A) =
∣∣∣∣∣∑
x∈LA
µ(x)
∣∣∣∣∣ .
Arrangements and Combinatorics – p. 35
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A corollary
Corollary. Let A be a real arrangement. Thenr(A) and b(A) depend only on L(A).
a
b
dc
a
b
c d
Arrangements and Combinatorics – p. 36
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A corollary
Corollary. Let A be a real arrangement. Thenr(A) and b(A) depend only on L(A).
a
b
dc
a
b
c d
Arrangements and Combinatorics – p. 36
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Faces
R(A): set of regions of A
Definition. A (closed) face of a realarrangement A is a set
∅ 6= F = R ∩ x,
where R ∈ R(A), x ∈ L(A), and R = closure ofR.
fk(A): number of k-dimensional faces (k-faces)of A
Arrangements and Combinatorics – p. 37
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Faces
R(A): set of regions of A
Definition. A (closed) face of a realarrangement A is a set
∅ 6= F = R ∩ x,
where R ∈ R(A), x ∈ L(A), and R = closure ofR.
fk(A): number of k-dimensional faces (k-faces)of A
Arrangements and Combinatorics – p. 37
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Example of fi(A)
f0(A) = 3, f1(A) = 9, f2(A) = r(A) = 7
Arrangements and Combinatorics – p. 38
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Formula for fk(A)
fk(A) =∑
x∈L(A)corank(x)=k
∑
y≥x
|µ(x, y)|
Proof. Easy consequence of Zaslavsky’s formulafor r(A). �
Arrangements and Combinatorics – p. 39
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Formula for fk(A)
fk(A) =∑
x∈L(A)corank(x)=k
∑
y≥x
|µ(x, y)|
Proof. Easy consequence of Zaslavsky’s formulafor r(A). �
Arrangements and Combinatorics – p. 39
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Zonotopes
Let X,Y ⊆ Kn
Minkowski sum:X + Y = {x + y : x ∈ X, y ∈ Y }
zonotope: a Minkowski sum L1 + · · · + Lk of linesegments in Rn
Arrangements and Combinatorics – p. 40
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Example of zonotope
(0,0)
(0,0)
Arrangements and Combinatorics – p. 41
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Example of zonotope
(0,0)
Arrangements and Combinatorics – p. 41
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Characterization of zonotopes
Theorem. Let P be a convex polytope. Thefollowing are equivalent.
P is a zonotope.
Every face of P is centrally-symmetric.
Every 2-dimensional face of P iscentrally-symmetric.
Arrangements and Combinatorics – p. 42
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The zonotope of a real arrangement
A: a real central arrangement
n1, . . . , nk: normals to H ∈ A
Li: line segment from 0 to ni
Z(A): the zonotope L1 + · · · + Lk
Arrangements and Combinatorics – p. 43
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Number of faces of Z(A)
Theorem. Let fi(Z(A)) denote the number ofi-dimensional faces of Z(A). Then
fi(Z(A)) = fn−i(A).
Informally, Z(A) is a “dual object” to A.
Arrangements and Combinatorics – p. 44
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Number of faces of Z(A)
Theorem. Let fi(Z(A)) denote the number ofi-dimensional faces of Z(A). Then
fi(Z(A)) = fn−i(A).
Informally, Z(A) is a “dual object” to A.
Arrangements and Combinatorics – p. 44
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An example of Z(A)
Arrangements and Combinatorics – p. 45
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An example of Z(A)
Arrangements and Combinatorics – p. 45
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Another example
hexagonal prism
Arrangements and Combinatorics – p. 46
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Another example
hexagonal prism
Arrangements and Combinatorics – p. 46
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Another example
rhombic dodecahedron
Arrangements and Combinatorics – p. 47
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Another example
rhombic dodecahedron
Arrangements and Combinatorics – p. 47
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Graphical arrangements
G: graph on vertex set [n] (no loops or multipleedges)
E(G): edge set of G
AG: arrangement in Kn with hyperplanes xi = xj
if ij ∈ E(G)
If G = Kn, the complete graph on [n], then AKn
is the braid arrangement Bn.
Arrangements and Combinatorics – p. 48
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Graphical arrangements
G: graph on vertex set [n] (no loops or multipleedges)
E(G): edge set of G
AG: arrangement in Kn with hyperplanes xi = xj
if ij ∈ E(G)
If G = Kn, the complete graph on [n], then AKn
is the braid arrangement Bn.
Arrangements and Combinatorics – p. 48
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Graphical arrangements
G: graph on vertex set [n] (no loops or multipleedges)
E(G): edge set of G
AG: arrangement in Kn with hyperplanes xi = xj
if ij ∈ E(G)
If G = Kn, the complete graph on [n], then AKn
is the braid arrangement Bn.
Arrangements and Combinatorics – p. 48
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Graphical arrangements
G: graph on vertex set [n] (no loops or multipleedges)
E(G): edge set of G
AG: arrangement in Kn with hyperplanes xi = xj
if ij ∈ E(G)
If G = Kn, the complete graph on [n], then AKn
is the braid arrangement Bn.
Arrangements and Combinatorics – p. 48
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Set partitions
partition of a finite set S: π = {B1, . . . , Bk}, suchthat
Bi 6= ∅,⋃
Bi = S, Bi ∩ Bj = ∅ (i 6= j)
Bi is a block of π.
ΠS: set of partitions of S
Let π, σ ∈ ΠS. Then π is a refinement of σ,written π ≤ σ, if every block of π is contained ina block of σ.
Arrangements and Combinatorics – p. 49
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The bond lattice of G
G: graph on vertex set [n]
connected partition of [n]: a partition of [n] forwhich each block induces a connected subgraphof G
bond lattice L(G) of G: set of connectedpartitions of [n], ordered by refinement
Arrangements and Combinatorics – p. 50
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The bond lattice of G
G: graph on vertex set [n]
connected partition of [n]: a partition of [n] forwhich each block induces a connected subgraphof G
bond lattice L(G) of G: set of connectedpartitions of [n], ordered by refinement
Arrangements and Combinatorics – p. 50
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Example of bond lattice
G
)L(G
12 13 23 14 34
123 124 134 234
1234
3412 14
23
4 3
21
Arrangements and Combinatorics – p. 51
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Bond lattices and intersection posets
G: graph with bond lattice L(G)
AG: graphical arrangement
Theorem. L(G) ∼= L(A(G))
Proof. Let Hij be the hyperplane defined byxi = xj, ij ∈ E(G). Let x ∈ L(A). Define verticesi ∼ j if x ⊆ Hij. Then ∼ is an equivalencerelation whose equivalence classes form aconnected partition of [n], etc. �
Arrangements and Combinatorics – p. 52
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Bond lattices and intersection posets
G: graph with bond lattice L(G)
AG: graphical arrangement
Theorem. L(G) ∼= L(A(G))
Proof. Let Hij be the hyperplane defined byxi = xj, ij ∈ E(G). Let x ∈ L(A). Define verticesi ∼ j if x ⊆ Hij. Then ∼ is an equivalencerelation whose equivalence classes form aconnected partition of [n], etc. �
Arrangements and Combinatorics – p. 52
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Chromatic polynomial of G
coloring of G is κ : [n] → P = {1, 2, . . . }
proper coloring: κ(i) 6= κ(j) if ij ∈ E(G)
chromatic polynomial: For q ≥ 1,
χG(q) = #{proper κ : [n] → [q]}
Easy fact: χG(q) ∈ Z[q]
Arrangements and Combinatorics – p. 53
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Chromatic polynomial of G
coloring of G is κ : [n] → P = {1, 2, . . . }
proper coloring: κ(i) 6= κ(j) if ij ∈ E(G)
chromatic polynomial: For q ≥ 1,
χG(q) = #{proper κ : [n] → [q]}
Easy fact: χG(q) ∈ Z[q]
Arrangements and Combinatorics – p. 53
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Chromatic polynomial of G
coloring of G is κ : [n] → P = {1, 2, . . . }
proper coloring: κ(i) 6= κ(j) if ij ∈ E(G)
chromatic polynomial: For q ≥ 1,
χG(q) = #{proper κ : [n] → [q]}
Easy fact: χG(q) ∈ Z[q]
Arrangements and Combinatorics – p. 53
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Chromatic polynomial of G
coloring of G is κ : [n] → P = {1, 2, . . . }
proper coloring: κ(i) 6= κ(j) if ij ∈ E(G)
chromatic polynomial: For q ≥ 1,
χG(q) = #{proper κ : [n] → [q]}
Easy fact: χG(q) ∈ Z[q]
Arrangements and Combinatorics – p. 53
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χA(G)(t)
Theorem. χA(G)(t) = χG(t)
Proof. Let σ ∈ L(G).
χσ(q) = number of f : [n] → [q] such that:
a, b in same block of σ ⇒ f(a) = f(b)
a, b in different blocks, ab ∈ E ⇒ f(a) 6= f(b).
Arrangements and Combinatorics – p. 54
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χA(G)(t)
Theorem. χA(G)(t) = χG(t)
Proof. Let σ ∈ L(G).
χσ(q) = number of f : [n] → [q] such that:
a, b in same block of σ ⇒ f(a) = f(b)
a, b in different blocks, ab ∈ E ⇒ f(a) 6= f(b).
Arrangements and Combinatorics – p. 54
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Continuation of proof
Given any f : [n] → [q], there is a uniqueσ ∈ L(G) such that f is enumerated by χσ(q).Hence ∀π ∈ L(G),
q#π =∑
σ≥π
χσ(q).
Möbius inversion ⇒ χπ(q) =∑
σ≥π
q#σµ(π, σ).
Note χ0̂(q) = χG(q). �
Arrangements and Combinatorics – p. 55
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Continuation of proof
Given any f : [n] → [q], there is a uniqueσ ∈ L(G) such that f is enumerated by χσ(q).Hence ∀π ∈ L(G),
q#π =∑
σ≥π
χσ(q).
Möbius inversion ⇒ χπ(q) =∑
σ≥π
q#σµ(π, σ).
Note χ0̂(q) = χG(q). �
Arrangements and Combinatorics – p. 55
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Continuation of proof
Given any f : [n] → [q], there is a uniqueσ ∈ L(G) such that f is enumerated by χσ(q).Hence ∀π ∈ L(G),
q#π =∑
σ≥π
χσ(q).
Möbius inversion ⇒ χπ(q) =∑
σ≥π
q#σµ(π, σ).
Note χ0̂(q) = χG(q). �
Arrangements and Combinatorics – p. 55
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Characteristic polynomial of Bn
Recall: Bn = A(Kn) (braid arrangement)
LBn
∼= Πn,
the lattice of all partitions of [n] (ordered byrefinement)
Clearly χKn(q) = q(q − 1) · · · (q − n + 1).
⇒ χBn(t) = t(t − 1) · · · (t − n + 1).
Arrangements and Combinatorics – p. 56
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Characteristic polynomial of Bn
Recall: Bn = A(Kn) (braid arrangement)
LBn
∼= Πn,
the lattice of all partitions of [n] (ordered byrefinement)
Clearly χKn(q) = q(q − 1) · · · (q − n + 1).
⇒ χBn(t) = t(t − 1) · · · (t − n + 1).
Arrangements and Combinatorics – p. 56
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Characteristic polynomial of Bn
Recall: Bn = A(Kn) (braid arrangement)
LBn
∼= Πn,
the lattice of all partitions of [n] (ordered byrefinement)
Clearly χKn(q) = q(q − 1) · · · (q − n + 1).
⇒ χBn(t) = t(t − 1) · · · (t − n + 1).
Arrangements and Combinatorics – p. 56
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Chordal graphs
A graph G is chordal (triangulated, rigidcircuit) if the vertices can be ordered v1, . . . , vn
so that for all i, vi is connected to a clique(complete subgraph) of the restriction of G to{v1, . . . , vi−1}.
Known fact: G is chordal if and only if everycycle of length at least four has a chord.
Arrangements and Combinatorics – p. 57
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Chordal graphs
A graph G is chordal (triangulated, rigidcircuit) if the vertices can be ordered v1, . . . , vn
so that for all i, vi is connected to a clique(complete subgraph) of the restriction of G to{v1, . . . , vi−1}.
Known fact: G is chordal if and only if everycycle of length at least four has a chord.
Arrangements and Combinatorics – p. 57
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Example of a chordal graph
7
2 5
1
4 6
83
Arrangements and Combinatorics – p. 58
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Chordal graph coloring
Let v1, . . . , vn be a vertex ordering so that for all i,vi is connected to a clique of the restriction Gi−1
of G to {v1, . . . , vi−1}.
Let ai be the number of vertices of Gi−1 to whichvi is connected (so a1 = 0). Once v1, . . . , vi−1 are(properly) colored, there are q − ai ways to colorvi. Hence
χG(q) = (q − a1)(q − a2) · · · (q − an).
Arrangements and Combinatorics – p. 59
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Chordal graph coloring
Let v1, . . . , vn be a vertex ordering so that for all i,vi is connected to a clique of the restriction Gi−1
of G to {v1, . . . , vi−1}.
Let ai be the number of vertices of Gi−1 to whichvi is connected (so a1 = 0). Once v1, . . . , vi−1 are(properly) colored, there are q − ai ways to colorvi. Hence
χG(q) = (q − a1)(q − a2) · · · (q − an).
Arrangements and Combinatorics – p. 59
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Chordal graph coloring
Let v1, . . . , vn be a vertex ordering so that for all i,vi is connected to a clique of the restriction Gi−1
of G to {v1, . . . , vi−1}.
Let ai be the number of vertices of Gi−1 to whichvi is connected (so a1 = 0). Once v1, . . . , vi−1 are(properly) colored, there are q − ai ways to colorvi. Hence
χG(q) = (q − a1)(q − a2) · · · (q − an).
Arrangements and Combinatorics – p. 59
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Acyclic orientations
Orientation of G: assignment o of a directioni → j or j → i to each edge.
Acyclic orientation: an orientation with nodirected cycles
Arrangements and Combinatorics – p. 60
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χG(−1)
Given o, define
Ro = {(x1, . . . , xn) ∈ Rn : xi < xj whenever i → j in o}.
Then Ro is a region of A(G), and conversely.(Conditions are consistent because o is acyclic.)
ao(G): number of acyclic orientations of G
Theorem. r(AG) = (−1)nχG(−1) = ao(G).
This proof is due to Greene (1977).
Arrangements and Combinatorics – p. 61
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χG(−1)
Given o, define
Ro = {(x1, . . . , xn) ∈ Rn : xi < xj whenever i → j in o}.
Then Ro is a region of A(G), and conversely.(Conditions are consistent because o is acyclic.)
ao(G): number of acyclic orientations of G
Theorem. r(AG) = (−1)nχG(−1) = ao(G).
This proof is due to Greene (1977).
Arrangements and Combinatorics – p. 61
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χG(−1)
Given o, define
Ro = {(x1, . . . , xn) ∈ Rn : xi < xj whenever i → j in o}.
Then Ro is a region of A(G), and conversely.(Conditions are consistent because o is acyclic.)
ao(G): number of acyclic orientations of G
Theorem. r(AG) = (−1)nχG(−1) = ao(G).
This proof is due to Greene (1977).
Arrangements and Combinatorics – p. 61
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χG(−1)
Given o, define
Ro = {(x1, . . . , xn) ∈ Rn : xi < xj whenever i → j in o}.
Then Ro is a region of A(G), and conversely.(Conditions are consistent because o is acyclic.)
ao(G): number of acyclic orientations of G
Theorem. r(AG) = (−1)nχG(−1) = ao(G).
This proof is due to Greene (1977).
Arrangements and Combinatorics – p. 61
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(−1)iµ(x, y)
Goal: interpret (−1)iµ(x, y) combinatorially,where i = rank(x, y).
For simplicity we deal only with hyperplanearrangements, though the “right” level ofgenerality is matroid theory.
Arrangements and Combinatorics – p. 62
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(−1)iµ(x, y)
Goal: interpret (−1)iµ(x, y) combinatorially,where i = rank(x, y).
For simplicity we deal only with hyperplanearrangements, though the “right” level ofgenerality is matroid theory.
Arrangements and Combinatorics – p. 62
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Broken circuits
A: central arrangement
circuit: a minimal linearly dependent subset of A
H1, H2, . . . , Hm: ordering of A
broken circuit: a set C − {H}, where C is acircuit and H the last element of C in the aboveordering
broken circuit complex:
BC(A) = {F ⊆ A : F contains no broken circuit}
Arrangements and Combinatorics – p. 63
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Broken circuits
A: central arrangement
circuit: a minimal linearly dependent subset of A
H1, H2, . . . , Hm: ordering of A
broken circuit: a set C − {H}, where C is acircuit and H the last element of C in the aboveordering
broken circuit complex:
BC(A) = {F ⊆ A : F contains no broken circuit}
Arrangements and Combinatorics – p. 63
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Broken circuits
A: central arrangement
circuit: a minimal linearly dependent subset of A
H1, H2, . . . , Hm: ordering of A
broken circuit: a set C − {H}, where C is acircuit and H the last element of C in the aboveordering
broken circuit complex:
BC(A) = {F ⊆ A : F contains no broken circuit}
Arrangements and Combinatorics – p. 63
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An example
Note: BC(A) is a simplicial complex, i.e.,F ∈ BC(A), G ⊆ F ⇒ G ∈ BC(A).
2 4
53
1
2
3
4
51
4
5
2
312
4 3
1
5
Arrangements and Combinatorics – p. 64
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An example
Note: BC(A) is a simplicial complex, i.e.,F ∈ BC(A), G ⊆ F ⇒ G ∈ BC(A).
2 4
53
1
2
3
4
51
4
5
2
312
4 3
1
5
Arrangements and Combinatorics – p. 64
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An example
Note: BC(A) is a simplicial complex, i.e.,F ∈ BC(A), G ⊆ F ⇒ G ∈ BC(A).
2 4
53
1
2
3
4
51
4
5
2
312
4 3
1
5
Arrangements and Combinatorics – p. 64
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Example (continued)
4
5
2
312
4 3
1
5
fi = fi(BC(A)): # i-dim. faces of BC(A)
f−1 = 1, f0 = 5, f1 = 8, f2 = 4
χA(t) = t3 − 5t2 + 8t − 4
Arrangements and Combinatorics – p. 65
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Example (continued)
4
5
2
312
4 3
1
5
fi = fi(BC(A)): # i-dim. faces of BC(A)
f−1 = 1, f0 = 5, f1 = 8, f2 = 4
χA(t) = t3 − 5t2 + 8t − 4
Arrangements and Combinatorics – p. 65
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Example (continued)
4
5
2
312
4 3
1
5
fi = fi(BC(A)): # i-dim. faces of BC(A)
f−1 = 1, f0 = 5, f1 = 8, f2 = 4
χA(t) = t3 − 5t2 + 8t − 4
Arrangements and Combinatorics – p. 65
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Covers
L = LA
y covers x in L: x < y, 6 ∃x < z < y
E(L): edges of Hasse diagram of L, i.e,
E(L) = {(x, y) : y covers x}
Arrangements and Combinatorics – p. 66
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Labelings
λ : E(L) → P is a labeling of L
If C : x = x0 < x1 < · · · < xk = y is a saturatedchain from x to y (i.e., each xi+1 covers xi),define
λ(C) = (λ(x0, x1), λ(x1, x2), . . . , λ(xk−1, xk))
C is increasing if
λ(x0, x1) ≤ λ(x1, x2) ≤ · · · ≤ λ(xk−1, xk).
Arrangements and Combinatorics – p. 67
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Labelings
λ : E(L) → P is a labeling of L
If C : x = x0 < x1 < · · · < xk = y is a saturatedchain from x to y (i.e., each xi+1 covers xi),define
λ(C) = (λ(x0, x1), λ(x1, x2), . . . , λ(xk−1, xk))
C is increasing if
λ(x0, x1) ≤ λ(x1, x2) ≤ · · · ≤ λ(xk−1, xk).
Arrangements and Combinatorics – p. 67
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Labelings
λ : E(L) → P is a labeling of L
If C : x = x0 < x1 < · · · < xk = y is a saturatedchain from x to y (i.e., each xi+1 covers xi),define
λ(C) = (λ(x0, x1), λ(x1, x2), . . . , λ(xk−1, xk))
C is increasing if
λ(x0, x1) ≤ λ(x1, x2) ≤ · · · ≤ λ(xk−1, xk).
Arrangements and Combinatorics – p. 67
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E-labelings
(a)
1 2 3
32131 2312
3 2 1
1
1 1
2
1
3
2 11
221
(b) (c)
E-labeling: a labeling for which every interval[x, y] has a unique increasing chain.
Arrangements and Combinatorics – p. 68
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E-labelings
(a)
1 2 3
32131 2312
3 2 1
1
1 1
2
1
3
2 11
221
(b) (c)
E-labeling: a labeling for which every interval[x, y] has a unique increasing chain.
Arrangements and Combinatorics – p. 68
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Labeling and Möbius functions
Theorem. Let λ be an E-labeling of L, and letx ≤ y in L, rank(x, y) = k. Then (−1)kµ(x, y) isequal to the number of strictly decreasingsaturated chains from x to y, i.e.,
(−1)kµ(x, y) = #{x = x0 < x1 < · · · < xk = y :
λ(x0, x1) > λ(x1, x2) > · · · > λ(xk−1, xk)}.
Arrangements and Combinatorics – p. 69
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Labeling L(A)
H1, . . . , Hm: ordering of A (as before)If y covers x in L(A) then define
λ̃(x, y) = max{i : x ∨ Hi = y}.
Arrangements and Combinatorics – p. 70
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Example of λ
1
2
3
4
5
5
1 2 3 4 5
32
4 5 45 5 4
2
5 54 4 25
1 2 3 4 5
21
3
1
Arrangements and Combinatorics – p. 71
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Properties of λ
Claim 1. Define λ : E(L(A)) → P by
λ(x, y) = m + 1 − λ̃(x, y).
Then λ is an E-labeling.
Claim 2. The broken circuit complex BC(M)
consists of all chain labels λ̃(C) (regarded as aset), where C is an increasing saturated chainfrom 0̂ to some x ∈ L(M). Moreover, all suchλ̃(C) are distinct.
Arrangements and Combinatorics – p. 72
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Properties of λ
Claim 1. Define λ : E(L(A)) → P by
λ(x, y) = m + 1 − λ̃(x, y).
Then λ is an E-labeling.
Claim 2. The broken circuit complex BC(M)
consists of all chain labels λ̃(C) (regarded as aset), where C is an increasing saturated chainfrom 0̂ to some x ∈ L(M). Moreover, all suchλ̃(C) are distinct.
Arrangements and Combinatorics – p. 72
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Example of Claim 2.
1
2
3
4
5
5
1 2 3 4 5
32
4 5 45 5 4
2
5 54 4 25
1 2 3 4 5
21
3
1
broken circuits : 12, 34, 124
BC(A) = {∅, 1, 2, 3, 4, 5, 13, 14, 15, 23, 24, 25, 35, 45,
135, 145, 235, 245}
Arrangements and Combinatorics – p. 73
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Example of Claim 2.
1
2
3
4
5
5
1 2 3 4 5
32
4 5 45 5 4
2
5 54 4 25
1 2 3 4 5
21
3
1
broken circuits : 12, 34, 124
BC(A) = {∅, 1, 2, 3, 4, 5, 13, 14, 15, 23, 24, 25, 35, 45,
135, 145, 235, 245}
Arrangements and Combinatorics – p. 73
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Broken circuit theorem
Immediate consequence of Claims 1 and 2:
Theorem. χA(t) =∑
F∈BC(A)
(−1)#F tn−#F
Corollary. The coefficients of χA(t) alternate insign, i.e., χA(t) = tn − a1t
n−1 + a2tn−2 − · · · ,
where ai ≥ 0. In fact
(−1)iµ(x, y) > 0, where i = rank(x, y).
Arrangements and Combinatorics – p. 74
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Broken circuit theorem
Immediate consequence of Claims 1 and 2:
Theorem. χA(t) =∑
F∈BC(A)
(−1)#F tn−#F
Corollary. The coefficients of χA(t) alternate insign, i.e., χA(t) = tn − a1t
n−1 + a2tn−2 − · · · ,
where ai ≥ 0. In fact
(−1)iµ(x, y) > 0, where i = rank(x, y).
Arrangements and Combinatorics – p. 74
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A glimpse of topology
[x, y]: (finite) interval in a poset P
ci: number of chains x = x0 < x1 < · · · < xi = y
Note. c0 = 0 unless x = y.
Philip Hall’s theorem (1936).µ(x, y) = c0 − c1 + c2 − · · ·
Arrangements and Combinatorics – p. 75
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A glimpse of topology
[x, y]: (finite) interval in a poset P
ci: number of chains x = x0 < x1 < · · · < xi = y
Note. c0 = 0 unless x = y.
Philip Hall’s theorem (1936).µ(x, y) = c0 − c1 + c2 − · · ·
Arrangements and Combinatorics – p. 75
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The order complex
P : a poset
order complex of P :
∆(P ) = {chains of P},
an abstract simplicial complex.
Write ∆(x, y) for the order complex of the openinterval (x, y) = {z ∈ P : x < z < y}.
Arrangements and Combinatorics – p. 76
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Example of an order complex
b
ca
f
x
a
c d
fe
b
y
d
e
P P)(∆Arrangements and Combinatorics – p. 77
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Euler characteristic
∆: finite simplicial complex
fi = # i-dimensional faces of ∆
Note: f−1 = 1 unless ∆ = ∅.
Euler characteristic: χ(∆) = f0 − f1 + f2 − · · ·
reduced Euler characteristic:χ̃(∆) = −f−1 + f0 − f1 + f2 − · · ·
Note: χ̃(∆) = χ(∆) − 1 unless ∆ = ∅.
Arrangements and Combinatorics – p. 78
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Euler characteristic
∆: finite simplicial complex
fi = # i-dimensional faces of ∆
Note: f−1 = 1 unless ∆ = ∅.
Euler characteristic: χ(∆) = f0 − f1 + f2 − · · ·
reduced Euler characteristic:χ̃(∆) = −f−1 + f0 − f1 + f2 − · · ·
Note: χ̃(∆) = χ(∆) − 1 unless ∆ = ∅.
Arrangements and Combinatorics – p. 78
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Euler characteristic
∆: finite simplicial complex
fi = # i-dimensional faces of ∆
Note: f−1 = 1 unless ∆ = ∅.
Euler characteristic: χ(∆) = f0 − f1 + f2 − · · ·
reduced Euler characteristic:χ̃(∆) = −f−1 + f0 − f1 + f2 − · · ·
Note: χ̃(∆) = χ(∆) − 1 unless ∆ = ∅.
Arrangements and Combinatorics – p. 78
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Philip Hall’s theorem restated
Theorem. For x < y in a finite poset,
µ(x, y) = χ̃(∆(x, y)).
Recall for any finite simplicial complex ∆,
χ̃(∆) =∑
j
(−1)j dim H̃j(∆; K),
where H̃j(∆; K) denotes reduced simplicialhomology over the field K.
Arrangements and Combinatorics – p. 79
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Philip Hall’s theorem restated
Theorem. For x < y in a finite poset,
µ(x, y) = χ̃(∆(x, y)).
Recall for any finite simplicial complex ∆,
χ̃(∆) =∑
j
(−1)j dim H̃j(∆; K),
where H̃j(∆; K) denotes reduced simplicialhomology over the field K.
Arrangements and Combinatorics – p. 79
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A topological question
For x < y in L(A), with i = rank(x, y), we have
d := dim ∆(x, y) = i − 2.
In particular, (−1)d = (−1)i.
We get:
d∑
j=0
(−1)d−j dim H̃j(∆; K) = (−1)iµ(x, y) > 0.
Is there a topological reason for this?
Arrangements and Combinatorics – p. 80
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A topological question
For x < y in L(A), with i = rank(x, y), we have
d := dim ∆(x, y) = i − 2.
In particular, (−1)d = (−1)i.
We get:
d∑
j=0
(−1)d−j dim H̃j(∆; K) = (−1)iµ(x, y) > 0.
Is there a topological reason for this?
Arrangements and Combinatorics – p. 80
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A topological question
For x < y in L(A), with i = rank(x, y), we have
d := dim ∆(x, y) = i − 2.
In particular, (−1)d = (−1)i.
We get:
d∑
j=0
(−1)d−j dim H̃j(∆; K) = (−1)iµ(x, y) > 0.
Is there a topological reason for this?
Arrangements and Combinatorics – p. 80
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Folkman’s theorem
Previous slide:d∑
j=0
(−1)d−j dim H̃j(∆; K) > 0.
Theorem (Folkman, 1966).
H̃j(∆; K)
{= 0, j 6= d
6= 0, j = d.
Note. dim H̃d(∆; K) = (−1)dµ(x, y)
Early result in topological combinatorics.
Arrangements and Combinatorics – p. 81
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Folkman’s theorem
Previous slide:d∑
j=0
(−1)d−j dim H̃j(∆; K) > 0.
Theorem (Folkman, 1966).
H̃j(∆; K)
{= 0, j 6= d
6= 0, j = d.
Note. dim H̃d(∆; K) = (−1)dµ(x, y)
Early result in topological combinatorics.
Arrangements and Combinatorics – p. 81
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Folkman’s theorem
Previous slide:d∑
j=0
(−1)d−j dim H̃j(∆; K) > 0.
Theorem (Folkman, 1966).
H̃j(∆; K)
{= 0, j 6= d
6= 0, j = d.
Note. dim H̃d(∆; K) = (−1)dµ(x, y)
Early result in topological combinatorics.
Arrangements and Combinatorics – p. 81
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Cohen-Macaulay posets
A finite poset P is Cohen-Macaulay (over K) ifafter adjoining a top and bottom element to P ,every interval [x, y] satisfies:
If d = dim ∆(x, y) then H̃j(∆(x, y); K) = 0whenever j 6= d.
Folkman’s theorem, restated. If A is centralthen L(A) is Cohen-Macaulay.
Arrangements and Combinatorics – p. 82
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Cohen-Macaulay posets
A finite poset P is Cohen-Macaulay (over K) ifafter adjoining a top and bottom element to P ,every interval [x, y] satisfies:
If d = dim ∆(x, y) then H̃j(∆(x, y); K) = 0whenever j 6= d.
Folkman’s theorem, restated. If A is centralthen L(A) is Cohen-Macaulay.
Arrangements and Combinatorics – p. 82
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Modular elements
Let A be central. An element x ∈ L(A) ismodular if for all y ∈ L we have
rk(x) + rk(y) = rk(x ∧ y) + rk(x ∨ y).
x y
x is not modular: rk(x) + rk(y) = 2 + 2 = 4,rk(x ∧ y) + rk(x ∨ y) = 0 + 3 = 3
Arrangements and Combinatorics – p. 83
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Modular elements
Let A be central. An element x ∈ L(A) ismodular if for all y ∈ L we have
rk(x) + rk(y) = rk(x ∧ y) + rk(x ∨ y).
x y
x is not modular: rk(x) + rk(y) = 2 + 2 = 4,rk(x ∧ y) + rk(x ∨ y) = 0 + 3 = 3
Arrangements and Combinatorics – p. 83
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Simple properties
Easy: 0̂ = Kn, 1̂ =⋂
H∈A H (the top element),and each H ∈ A is modular.
Arrangements and Combinatorics – p. 84
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More properties
x, y ∈ L(A) are complements if x ∧ y = 0̂,x ∨ y = 1̂.
Theorem. Let r = rk(A). Let x ∈ L. Thefollowing four conditions are equivalent.
(i) x is a modular element of L.
(ii) If x ∧ y = 0̂, then rk(x) + rk(y) = rk(x ∨ y).
(iii) If x and y are complements, thenrk(x) + rk(y) = n.
(iv) All complements of x are incomparable.
Arrangements and Combinatorics – p. 85
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More properties
x, y ∈ L(A) are complements if x ∧ y = 0̂,x ∨ y = 1̂.
Theorem. Let r = rk(A). Let x ∈ L. Thefollowing four conditions are equivalent.
(i) x is a modular element of L.
(ii) If x ∧ y = 0̂, then rk(x) + rk(y) = rk(x ∨ y).
(iii) If x and y are complements, thenrk(x) + rk(y) = n.
(iv) All complements of x are incomparable.
Arrangements and Combinatorics – p. 85
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Two additional results
Theorem.
(a) (transitivity of modularity) If x is a modularelement of L and y is modular in the interval[0̂, x], then y is a modular element of L.
(b) If x and y are modular elements of L, thenx ∧ y is also modular.
Arrangements and Combinatorics – p. 86
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Modular element factorization thm.
Theorem. Let z be a modular element of L(A),A central of rank r. Write χz(t) = χ[0̂,z](t). Then
χL(t) = χz(t)
∑
y : y∧z=0̂
µL(y)tn−rk(y)−rk(z)
.
Since each H ∈ A is modular in L(A), we get:
Corollary. For all H ∈ A,
χL(t) = (t − 1)∑
y∧H=0̂
µ(y)tn−1−rk(y).
Arrangements and Combinatorics – p. 87
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Modular element factorization thm.
Theorem. Let z be a modular element of L(A),A central of rank r. Write χz(t) = χ[0̂,z](t). Then
χL(t) = χz(t)
∑
y : y∧z=0̂
µL(y)tn−rk(y)−rk(z)
.
Since each H ∈ A is modular in L(A), we get:
Corollary. For all H ∈ A,
χL(t) = (t − 1)∑
y∧H=0̂
µ(y)tn−1−rk(y).
Arrangements and Combinatorics – p. 87
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Supersolvability
A central arrangement A (or L(A)) issupersolvable if L(A) has a maximal chain0̂ = x0 < x1 < · · · < xr = 1̂ of modular elementsxi.
In this case, letai = #{H ∈ A : H ≤ xi, H 6≤ xi−1}.
Corollary. If A is supersolvable, then
χA(t) = tn−r(t − a1)(t − a2) · · · (t − ar).
Arrangements and Combinatorics – p. 88
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Supersolvability
A central arrangement A (or L(A)) issupersolvable if L(A) has a maximal chain0̂ = x0 < x1 < · · · < xr = 1̂ of modular elementsxi.
In this case, letai = #{H ∈ A : H ≤ xi, H 6≤ xi−1}.
Corollary. If A is supersolvable, then
χA(t) = tn−r(t − a1)(t − a2) · · · (t − ar).
Arrangements and Combinatorics – p. 88
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Chordal graphs, revisited
For what graphs G is AG supersolvable?
Recall: xi = xj for ij ∈ E(G)
Recall that a chordal graph has a vertexordering v1, . . . , vn so that for all i, vi is connectedto a clique of the restriction Gi−1 of G to{v1, . . . , vi−1}.
If vi is connected to ai vertices of Gi−1, then
χG(q) = (q − a1)(q − a2) · · · (q − an).
Arrangements and Combinatorics – p. 89
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Chordal graphs, revisited
For what graphs G is AG supersolvable?
Recall: xi = xj for ij ∈ E(G)
Recall that a chordal graph has a vertexordering v1, . . . , vn so that for all i, vi is connectedto a clique of the restriction Gi−1 of G to{v1, . . . , vi−1}.
If vi is connected to ai vertices of Gi−1, then
χG(q) = (q − a1)(q − a2) · · · (q − an).
Arrangements and Combinatorics – p. 89
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Chordal graphs, revisited
For what graphs G is AG supersolvable?
Recall: xi = xj for ij ∈ E(G)
Recall that a chordal graph has a vertexordering v1, . . . , vn so that for all i, vi is connectedto a clique of the restriction Gi−1 of G to{v1, . . . , vi−1}.
If vi is connected to ai vertices of Gi−1, then
χG(q) = (q − a1)(q − a2) · · · (q − an).
Arrangements and Combinatorics – p. 89
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Supersolvable graphs
Suggests that
G chordal ⇒ G (or AG) supersolvable.
In fact:
Theorem. G is chordal if and only if AG issupersolvable.
Arrangements and Combinatorics – p. 90
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Supersolvable graphs
Suggests that
G chordal ⇒ G (or AG) supersolvable.
In fact:
Theorem. G is chordal if and only if AG issupersolvable.
Arrangements and Combinatorics – p. 90
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Free arrangements
Saito defined free arrangements A. Terao(1980) proved
χA(t) = (t − a1) · · · (t − an),
where ai ∈ {0, 1, 2, . . . }. (Definition not givenhere.)
Supersolvable arrangements are free.
Open: is freeness of A a combinatorialproperty? That is, does it just depend on χA(t)?
Arrangements and Combinatorics – p. 91
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Free arrangements
Saito defined free arrangements A. Terao(1980) proved
χA(t) = (t − a1) · · · (t − an),
where ai ∈ {0, 1, 2, . . . }. (Definition not givenhere.)
Supersolvable arrangements are free.
Open: is freeness of A a combinatorialproperty? That is, does it just depend on χA(t)?
Arrangements and Combinatorics – p. 91
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Free arrangements
Saito defined free arrangements A. Terao(1980) proved
χA(t) = (t − a1) · · · (t − an),
where ai ∈ {0, 1, 2, . . . }. (Definition not givenhere.)
Supersolvable arrangements are free.
Open: is freeness of A a combinatorialproperty? That is, does it just depend on χA(t)?
Arrangements and Combinatorics – p. 91
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Finite fields and good reduction
A: arrangement over Q
By multiplying hyperplane equations by asuitable integer, can assume A is defined over Z.
Consider coefficients modulo a prime p to get anarrangment Aq defined over the finite field Fq,q = pk.
Aq has good reduction if LA∼= LAq
.
Arrangements and Combinatorics – p. 92
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Finite fields and good reduction
A: arrangement over Q
By multiplying hyperplane equations by asuitable integer, can assume A is defined over Z.
Consider coefficients modulo a prime p to get anarrangment Aq defined over the finite field Fq,q = pk.
Aq has good reduction if LA∼= LAq
.
Arrangements and Combinatorics – p. 92
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Finite fields and good reduction
A: arrangement over Q
By multiplying hyperplane equations by asuitable integer, can assume A is defined over Z.
Consider coefficients modulo a prime p to get anarrangment Aq defined over the finite field Fq,q = pk.
Aq has good reduction if LA∼= LAq
.
Arrangements and Combinatorics – p. 92
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Almost always good reduction
Example. A = {2, 10}: affine arrangement inQ1 = Q. Good reduction ⇔ p 6= 2, 5.
Theorem. Let A be an arrangement defined overZ. Then A has good reduction for all but finitelymany primes p.
Proof idea. Consider minors of the coefficientmatrix, etc. �
Arrangements and Combinatorics – p. 93
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Almost always good reduction
Example. A = {2, 10}: affine arrangement inQ1 = Q. Good reduction ⇔ p 6= 2, 5.
Theorem. Let A be an arrangement defined overZ. Then A has good reduction for all but finitelymany primes p.
Proof idea. Consider minors of the coefficientmatrix, etc. �
Arrangements and Combinatorics – p. 93
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Almost always good reduction
Example. A = {2, 10}: affine arrangement inQ1 = Q. Good reduction ⇔ p 6= 2, 5.
Theorem. Let A be an arrangement defined overZ. Then A has good reduction for all but finitelymany primes p.
Proof idea. Consider minors of the coefficientmatrix, etc. �
Arrangements and Combinatorics – p. 93
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The finite field method
Theorem. Let A be an arrangement in Qn, andsuppose that L(A) ∼= L(Aq) for some primepower q. Then
χA(q) = #
Fn
q −⋃
H∈Aq
H
= qn − #⋃
H∈Aq
H.
Arrangements and Combinatorics – p. 94
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Proof
Let x ∈ L(Aq) so #x = qdim(x) (computed eitherover Q or Fq). Define f, g : L(Aq) → Z by
f(x) = #x
g(x) = #
(x −
⋃
y>x
y
)
⇒ g(0̂) = g(Fnq ) = #
Fn
q −⋃
H∈Aq
H
.
Arrangements and Combinatorics – p. 95
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Proof concluded
Clearly f(x) =∑
y≥x
g(y).
Möbius inversion ⇒
g(x) =∑
y≥x
µ(x, y)f(y)
=∑
y≥x
µ(x, y)qdim(y)
x = 0̂ ⇒ g(0̂) =∑
y
µ(y)qdim(y) = χA(q) �
Arrangements and Combinatorics – p. 96
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Proof concluded
Clearly f(x) =∑
y≥x
g(y).
Möbius inversion ⇒
g(x) =∑
y≥x
µ(x, y)f(y)
=∑
y≥x
µ(x, y)qdim(y)
x = 0̂ ⇒ g(0̂) =∑
y
µ(y)qdim(y) = χA(q) �
Arrangements and Combinatorics – p. 96
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Proof concluded
Clearly f(x) =∑
y≥x
g(y).
Möbius inversion ⇒
g(x) =∑
y≥x
µ(x, y)f(y)
=∑
y≥x
µ(x, y)qdim(y)
x = 0̂ ⇒ g(0̂) =∑
y
µ(y)qdim(y) = χA(q) �
Arrangements and Combinatorics – p. 96
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Graphical arrangements
G: graph on vertex set 1, 2, . . . , n
AG: graphical arrangement xi = xj, ij ∈ E(G)
finite field method: for p >> 0 (actually, all p),
χAG
(q) = #{(α1, . . . , αn) ∈ Fnq : αi 6= αj if ij ∈ E(G)}
= χG(q)
!
Arrangements and Combinatorics – p. 97
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Graphical arrangements
G: graph on vertex set 1, 2, . . . , n
AG: graphical arrangement xi = xj, ij ∈ E(G)
finite field method: for p >> 0 (actually, all p),
χAG
(q) = #{(α1, . . . , αn) ∈ Fnq : αi 6= αj if ij ∈ E(G)}
= χG(q)
!
Arrangements and Combinatorics – p. 97
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Graphical arrangements
G: graph on vertex set 1, 2, . . . , n
AG: graphical arrangement xi = xj, ij ∈ E(G)
finite field method: for p >> 0 (actually, all p),
χAG
(q) = #{(α1, . . . , αn) ∈ Fnq : αi 6= αj if ij ∈ E(G)}
= χG(q)
!
Arrangements and Combinatorics – p. 97
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Graphical arrangements
G: graph on vertex set 1, 2, . . . , n
AG: graphical arrangement xi = xj, ij ∈ E(G)
finite field method: for p >> 0 (actually, all p),
χAG
(q) = #{(α1, . . . , αn) ∈ Fnq : αi 6= αj if ij ∈ E(G)}
= χG(q)
!
Arrangements and Combinatorics – p. 97
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The braid arrangement B(Bn)
xi − xj = 0, 1 ≤ i < j ≤ n
xi + xj = 0, 1 ≤ i < j ≤ n
xi = 0, 1 ≤ i ≤ n
Thus for p >> 0 (actually p > 2),
χB(Bn)(q) = #{(α1, . . . , αn) ∈ Fnq :
αi 6= ±αj (i 6= j), αi 6= 0}.
Choose α1 in q − 1 ways, then α2 in q − 3 ways,etc.
Arrangements and Combinatorics – p. 98
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The braid arrangement B(Bn)
xi − xj = 0, 1 ≤ i < j ≤ n
xi + xj = 0, 1 ≤ i < j ≤ n
xi = 0, 1 ≤ i ≤ n
Thus for p >> 0 (actually p > 2),
χB(Bn)(q) = #{(α1, . . . , αn) ∈ Fnq :
αi 6= ±αj (i 6= j), αi 6= 0}.
Choose α1 in q − 1 ways, then α2 in q − 3 ways,etc.
Arrangements and Combinatorics – p. 98
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The braid arrangement B(Bn)
xi − xj = 0, 1 ≤ i < j ≤ n
xi + xj = 0, 1 ≤ i < j ≤ n
xi = 0, 1 ≤ i ≤ n
Thus for p >> 0 (actually p > 2),
χB(Bn)(q) = #{(α1, . . . , αn) ∈ Fnq :
αi 6= ±αj (i 6= j), αi 6= 0}.
Choose α1 in q − 1 ways, then α2 in q − 3 ways,etc.
Arrangements and Combinatorics – p. 98
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Characteristic polynomial of B(Bn)
⇒ χB(Bn)(q) = (q − 1)(q − 3) · · · (q − 2n + 1)
In fact, B(Bn) is supersolvable.
Arrangements and Combinatorics – p. 99
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Characteristic polynomial of B(Bn)
⇒ χB(Bn)(q) = (q − 1)(q − 3) · · · (q − 2n + 1)
In fact, B(Bn) is supersolvable.
Arrangements and Combinatorics – p. 99
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B(Dn)
xi − xj = 0, 1 ≤ i < j ≤ n
xi + xj = 0, 1 ≤ i < j ≤ n
Exercise: If n ≥ 3 then
χB(Dn) = (q − 1)(q − 3) · · · (q − 2n + 3) · (q − n + 1).
Not supersolvable (n ≥ 4), but it is free.
Arrangements and Combinatorics – p. 100
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B(Dn)
xi − xj = 0, 1 ≤ i < j ≤ n
xi + xj = 0, 1 ≤ i < j ≤ n
Exercise: If n ≥ 3 then
χB(Dn) = (q − 1)(q − 3) · · · (q − 2n + 3) · (q − n + 1).
Not supersolvable (n ≥ 4), but it is free.
Arrangements and Combinatorics – p. 100
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B(Dn)
xi − xj = 0, 1 ≤ i < j ≤ n
xi + xj = 0, 1 ≤ i < j ≤ n
Exercise: If n ≥ 3 then
χB(Dn) = (q − 1)(q − 3) · · · (q − 2n + 3) · (q − n + 1).
Not supersolvable (n ≥ 4), but it is free.
Arrangements and Combinatorics – p. 100
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The Shi arrangement
Sn : xi − xj = 0, 1, 1 ≤ i < j ≤ n
dimSn = n, rkSn = n − 1, #Sn = n(n − 1)
Arrangements and Combinatorics – p. 101
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The Shi arrangement
Sn : xi − xj = 0, 1, 1 ≤ i < j ≤ n
dimSn = n, rkSn = n − 1, #Sn = n(n − 1)
Arrangements and Combinatorics – p. 101
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Characteristic polynomial of Sn
Theorem. χSn(t) = t(t − n)n−1, so
r(Sn) = (n + 1)n−1, b(Sn) = (n − 1)n−1.
Proof. Finite field method ⇒
χSn(p) = #{(α1, . . . , αn) ∈ Fn
p :
i < j ⇒ αi 6= αj and αi 6= αj + 1},
for p >> 0 (actually, all p).
Arrangements and Combinatorics – p. 102
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Characteristic polynomial of Sn
Theorem. χSn(t) = t(t − n)n−1, so
r(Sn) = (n + 1)n−1, b(Sn) = (n − 1)n−1.
Proof. Finite field method ⇒
χSn(p) = #{(α1, . . . , αn) ∈ Fn
p :
i < j ⇒ αi 6= αj and αi 6= αj + 1},
for p >> 0 (actually, all p).
Arrangements and Combinatorics – p. 102
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Proof continued
Choose π = (B1, . . . , Bp−n) such that⋃
Bi = [n], Bi ∩ Bj = ∅ if i 6= j, 1 ∈ B1.
For 2 ≤ k ≤ n there are p − n choices for i suchthat k ∈ Bi, so (p − n)n−1 choices in all.
Arrangements and Combinatorics – p. 103
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Proof continued
Choose π = (B1, . . . , Bp−n) such that⋃
Bi = [n], Bi ∩ Bj = ∅ if i 6= j, 1 ∈ B1.
For 2 ≤ k ≤ n there are p − n choices for i suchthat k ∈ Bi, so (p − n)n−1 choices in all.
Arrangements and Combinatorics – p. 103
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Circular placement of Fp
Arrange the elements of Fp clockwise on a circle.
Place 1, 2, . . . , n on some n of these points asfollows.
Place elements of B1 consecutively (clockwise)in increasing order with 1 placed at someelement α1 ∈ Fp.
Skip a space and place the elements of B2
consecutively in increasing order.
Skip another space and place the elements of B3
consecutively in increasing order, etc.
Arrangements and Combinatorics – p. 104
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Example for p = 11, n = 6
π = ({1, 4}, {5}, ∅, {2, 3, 6}, ∅)
0
1
2
3
45
6
7
8
9
102
6
3
1
4
5
Arrangements and Combinatorics – p. 105
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Conclusion of proof
αi: position (element of Fp) at which i was placed
Previous example:(α1, . . . , α6) = (6, 1, 2, 7, 9, 3) ∈ F6
11
Gives bijection
{(π = (B1, . . . , Bp−n), α1)} → Fnp −
⋃
H∈(Sn)p
H.
(p − n)n−1 choices for π and p choices for α1, so
χSn(p) = p(p − n)n−1.
Arrangements and Combinatorics – p. 106
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Conclusion of proof
αi: position (element of Fp) at which i was placed
Previous example:(α1, . . . , α6) = (6, 1, 2, 7, 9, 3) ∈ F6
11
Gives bijection
{(π = (B1, . . . , Bp−n), α1)} → Fnp −
⋃
H∈(Sn)p
H.
(p − n)n−1 choices for π and p choices for α1, so
χSn(p) = p(p − n)n−1.
Arrangements and Combinatorics – p. 106
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Conclusion of proof
αi: position (element of Fp) at which i was placed
Previous example:(α1, . . . , α6) = (6, 1, 2, 7, 9, 3) ∈ F6
11
Gives bijection
{(π = (B1, . . . , Bp−n), α1)} → Fnp −
⋃
H∈(Sn)p
H.
(p − n)n−1 choices for π and p choices for α1, so
χSn(p) = p(p − n)n−1.
Arrangements and Combinatorics – p. 106
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Conclusion of proof
αi: position (element of Fp) at which i was placed
Previous example:(α1, . . . , α6) = (6, 1, 2, 7, 9, 3) ∈ F6
11
Gives bijection
{(π = (B1, . . . , Bp−n), α1)} → Fnp −
⋃
H∈(Sn)p
H.
(p − n)n−1 choices for π and p choices for α1, so
χSn(p) = p(p − n)n−1.
Arrangements and Combinatorics – p. 106
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The Catalan arrangement
Cn : xi − xj = 0,−1, 1, 1 ≤ i < j ≤ n
dim Cn = n, rk Cn = n − 1, #Sn = 3
(n
2
)
Arrangements and Combinatorics – p. 107
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The Catalan arrangement
Cn : xi − xj = 0,−1, 1, 1 ≤ i < j ≤ n
dim Cn = n, rk Cn = n − 1, #Sn = 3
(n
2
)
Arrangements and Combinatorics – p. 107
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Char. poly. of Catalan arrangment
Theorem.χ
Cn(t) = t(t−n−1)(t−n−2)(t−n−3) · · · (t−2n+1),
sor(Cn) = n!Cn, b(Cn) = n!Cn−1,
where Cm = 1m+1
(2mm
)(Catalan number).
Easy to prove using finite field method.
Each region of the braid arrangement Bn
contains Cn regions and Cn−1 relatively boundedregions of the Catalan arrangment Cn.
Arrangements and Combinatorics – p. 108
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Char. poly. of Catalan arrangment
Theorem.χ
Cn(t) = t(t−n−1)(t−n−2)(t−n−3) · · · (t−2n+1),
sor(Cn) = n!Cn, b(Cn) = n!Cn−1,
where Cm = 1m+1
(2mm
)(Catalan number).
Easy to prove using finite field method.
Each region of the braid arrangement Bn
contains Cn regions and Cn−1 relatively boundedregions of the Catalan arrangment Cn.
Arrangements and Combinatorics – p. 108
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Char. poly. of Catalan arrangment
Theorem.χ
Cn(t) = t(t−n−1)(t−n−2)(t−n−3) · · · (t−2n+1),
sor(Cn) = n!Cn, b(Cn) = n!Cn−1,
where Cm = 1m+1
(2mm
)(Catalan number).
Easy to prove using finite field method.
Each region of the braid arrangement Bn
contains Cn regions and Cn−1 relatively boundedregions of the Catalan arrangment Cn.
Arrangements and Combinatorics – p. 108
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Catalan numbers
≥ 172 combinatorial interpretations of Cn at
math.mit.edu/∼rstan/ec
Arrangements and Combinatorics – p. 109
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The Linial arrangement
Ln : xi − xj = 1, 1 ≤ i < j ≤ n
dimLn = n, rkLn = n − 1, #Ln =
(n
2
)
Arrangements and Combinatorics – p. 110
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The Linial arrangement
Ln : xi − xj = 1, 1 ≤ i < j ≤ n
dimLn = n, rkLn = n − 1, #Ln =
(n
2
)
Arrangements and Combinatorics – p. 110
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Char. poly. of Linial arrangment
Theorem. χLn
(t) =t
2n
n∑
k=1
(n
k
)(t − k)n−1,
so
r(Ln) =1
2n
n∑
k=1
(n
k
)(k + 1)n−1
b(Ln) =1
2n
n∑
k=1
(n
k
)(k − 1)n−1
Arrangements and Combinatorics – p. 111
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Two proofs
Postnikov: (difficult) proof using Whitney’stheorem
Athanasiadis: (difficult) proof using finite fieldmethod
Arrangements and Combinatorics – p. 112
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Alternating trees
An alternating tree on [n] is a tree on the vertexset [n] such that every vertex is either less thanall its neighbors or greater than all its neighbors.
1 3 2 4
1 4 2 3
2 1 4 3
2 3 1 4
2 4 1 3
4
1 2
3
32
1
4
Arrangements and Combinatorics – p. 113
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Alternating trees
An alternating tree on [n] is a tree on the vertexset [n] such that every vertex is either less thanall its neighbors or greater than all its neighbors.
1 3 2 4
1 4 2 3
2 1 4 3
2 3 1 4
2 4 1 3
4
1 2
3
32
1
4
Arrangements and Combinatorics – p. 113
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Alternating trees and Ln
f(n): number of alternating trees on [n]
Theorem (Kuznetsov, Pak, Postnikov, 1994).
f(n + 1) =1
2n
n∑
k=1
(n
k
)(k + 1)n−1
Corollary. f(n + 1) = r(Ln)
No combinatorial proof known!
Arrangements and Combinatorics – p. 114
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Alternating trees and Ln
f(n): number of alternating trees on [n]
Theorem (Kuznetsov, Pak, Postnikov, 1994).
f(n + 1) =1
2n
n∑
k=1
(n
k
)(k + 1)n−1
Corollary. f(n + 1) = r(Ln)
No combinatorial proof known!
Arrangements and Combinatorics – p. 114
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Alternating trees and Ln
f(n): number of alternating trees on [n]
Theorem (Kuznetsov, Pak, Postnikov, 1994).
f(n + 1) =1
2n
n∑
k=1
(n
k
)(k + 1)n−1
Corollary. f(n + 1) = r(Ln)
No combinatorial proof known!
Arrangements and Combinatorics – p. 114
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The threshold arrangment
Tn : xi + xj = 0, 1 ≤ i < j ≤ n
dim Tn = n, rk Tn = n, #Tn =
(n
2
)
threshold graph:
∅ is a threshold graph
G threshhold ⇒ G ∪ {vertex} threshold
G threshold ⇒ join(G, v) threshold
Arrangements and Combinatorics – p. 115
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The threshold arrangment
Tn : xi + xj = 0, 1 ≤ i < j ≤ n
dim Tn = n, rk Tn = n, #Tn =
(n
2
)
threshold graph:
∅ is a threshold graph
G threshhold ⇒ G ∪ {vertex} threshold
G threshold ⇒ join(G, v) threshold
Arrangements and Combinatorics – p. 115
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Char. poly. of threshold arrangement
Theorem. r(Tn) = # threshold graphs on [n].Hence (by a known result on threshold graphs)
∑
n≥0
r(Tn)xn
n!=
ex(1 − x)
2 − ex.
Theorem.∑
n≥0
χTn
(t)xn
n!= (1 + x)(2ex − 1)(t−1)/2
Arrangements and Combinatorics – p. 116
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Char. poly. of threshold arrangement
Theorem. r(Tn) = # threshold graphs on [n].Hence (by a known result on threshold graphs)
∑
n≥0
r(Tn)xn
n!=
ex(1 − x)
2 − ex.
Theorem.∑
n≥0
χTn
(t)xn
n!= (1 + x)(2ex − 1)(t−1)/2
Arrangements and Combinatorics – p. 116
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Small values of χTn(t)
χT3(t) = t3 − 3t2 + 3t − 1
χT4(t) = t4 − 6t3 + 15t2 − 17t + 7
χT5(t) = t5 − 10t4 + 45t3 − 105t2 + 120t − 51.
Arrangements and Combinatorics – p. 117
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Coefficients of χTn(t)
Let
χTn(t) = tn − an−1t
n−1 + · · · + (−1)na0.
Thus∑
ai = #{threshold graphs on [n]}.
Open: interpret ai as the number of thresholdgraphs on [n] with some property.
Arrangements and Combinatorics – p. 118
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Coefficients of χTn(t)
Let
χTn(t) = tn − an−1t
n−1 + · · · + (−1)na0.
Thus∑
ai = #{threshold graphs on [n]}.
Open: interpret ai as the number of thresholdgraphs on [n] with some property.
Arrangements and Combinatorics – p. 118
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Coefficients of χTn(t)
Let
χTn(t) = tn − an−1t
n−1 + · · · + (−1)na0.
Thus∑
ai = #{threshold graphs on [n]}.
Open: interpret ai as the number of thresholdgraphs on [n] with some property.
Arrangements and Combinatorics – p. 118
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Minkowski space R1,3
R1,3: Minkowski spacetime with one time andthree space dimensions
p = (t,x) ∈ R1,3, x = (x, y, z) ∈ R3
|p|2 = t2 − |x|2 = t2 − (x2 + y2 + z2)
Arrangements and Combinatorics – p. 119
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Ordering events in R1,3
Let p1, . . . ,pk ∈ R1,3. In different referenceframes (at constant velocities with respect toeach other) these events can occur in differentorders (but never violating causality).
Main question: what is the maximum number ofdifferent orders in which these events can occur?
Arrangements and Combinatorics – p. 120
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Ordering events in R1,3
Let p1, . . . ,pk ∈ R1,3. In different referenceframes (at constant velocities with respect toeach other) these events can occur in differentorders (but never violating causality).
Main question: what is the maximum number ofdifferent orders in which these events can occur?
Arrangements and Combinatorics – p. 120
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The hyperplane of simultaneity
Let p1 = (t1,x1), p2 = (t2,x2) ∈ R1,3.
For a reference frame at velocity v, the Lorentztransformation ⇒ p1,p2 occur at the same time ifand only if
t1 − t2 = (x1 − x2) · v.
The set of all such v ∈ R3 forms a hyperplane.
Arrangements and Combinatorics – p. 121
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The Einstein arrangement
Thus the number of different orders in which theevents can occur is the number of regions R ofthe Einstein arrangement
E = E(p1, . . . ,pk)
defined by
ti − tj = (x1 − x2) · v, 1 ≤ i < j ≤ k,
such that |v| < 1 (the speed of light) for somev ∈ R.
Arrangements and Combinatorics – p. 122
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Intersection poset of E
Can insure that v ∈ R for all R by takingp1, . . . ,pk sufficiently “far apart”.
Can maximize r(E) for fixed k by choosingp1, . . . ,pk generic.
In this case, L(E) is isomorphic to the rank 3truncation of L(Bk) ∼= Πk.
Arrangements and Combinatorics – p. 123
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Intersection poset of E
Can insure that v ∈ R for all R by takingp1, . . . ,pk sufficiently “far apart”.
Can maximize r(E) for fixed k by choosingp1, . . . ,pk generic.
In this case, L(E) is isomorphic to the rank 3truncation of L(Bk) ∼= Πk.
Arrangements and Combinatorics – p. 123
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Coefficients of χBk(t)
Recall
χBk(t) = t(t − 1) · · · (t − k + 1)
= c(k, k)tk − c(k, k − 1)tk−1 + · · · ,
where c(k, i) is the number of permutations of1, 2, . . . , k with i cycles (signless Stirling numberof the first kind).
Arrangements and Combinatorics – p. 124
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Computation of r(E)
Corollary.
χE(t) = c(k, k)t3 − c(k, k − 1)t2 + c(k, k − 2)t
−c(k, k − 3)
⇒ r(E) = c(k, k) + c(k, k − 1) + c(k, k − 2)
+c(k, k − 3)
=1
48
(k6 − 7k5 + 23k4 − 37k3 + 48k2
−28k + 48)
Arrangements and Combinatorics – p. 125
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Arrangements and Combinatorics – p. 126