SLIDE 1 The top
the p ermutation pattern p
Eina r Steingrmsson Universit y
Strath lyde W
b y Jason P . Smith and joint w
with P eter M Nama ra and with A. Burstein, V. Jelnek and E. Jelnk
SLIDE 2 An
a pattern p in a p ermutation π is a sub- sequen e in π whose letters app ea r in the same
size as those in p. 463 is an
231 in 416325 1
SLIDE 3 An
a pattern p in a p ermutation π is a sub- sequen e in π whose letters app ea r in the same
size as those in p. 463 is an
231 in 416325 2
SLIDE 4 An
a pattern p in a p ermutation π is a sub- sequen e in π whose letters app ea r in the same
size as those in p. 463 is an
231 in 416325 If π has no
then π avoids p. 4173625 avoids 4321 3
SLIDE 5 An
a pattern p in a p ermutation π is a sub- sequen e in π whose letters app ea r in the same
size as those in p. 463 is an
231 in 416325 If π has no
then π avoids p. 4173625 avoids 4321 (No de reasing subsequen e
length 4) 4
SLIDE 6 The set
all p ermutations fo rms a p
with resp e t to pattern
5
SLIDE 7 The set
all p ermutations fo rms a p
with resp e t to pattern
σ τ
if σ is a pattern in τ 6
SLIDE 8 · · · 2314 · · ·
123 132 213 231 312 321 12 21 1 The b
the p
7
SLIDE 9 · · · 2314 · · ·
123 132 213 231 312 321 12 21 1 The b
the p
The b
the p
2314
213 as a pattern 8
SLIDE 10
· · · 2314 · · ·
123 132 213 231 312 321 12 21 1 The interval [12, 2314] = {π | 12 π 2314} 9
SLIDE 11
· · · 2314 · · ·
123 132 213 231 312 321 12 21 1 The interval [12, 2314] = {π | 12 π 2314} 10
SLIDE 12 2314 231 3142 1342 2413 3421 24135 23154 34215 24153 31524 42153 13524 35214 352164 352146 421635 241635 342165 3521746
11
SLIDE 13 The Mbius fun tion
an interval I
Mbius fun tion
is dened b y µ(x, x) = 1 and
if x < y 12
SLIDE 14 Computing µ(•, y)
an interval I
1 2
1
Mbius fun tion
is dened b y µ(x, x) = 1 and
if x < y 13
SLIDE 15 Computing the Mbius fun tion fo r the pattern p
A very sho rt p rehisto ry 14
SLIDE 16 Computing the Mbius fun tion fo r the pattern p
A very sho rt p rehisto ry Wilf (2002): Should b e done 15
SLIDE 17 Computing the Mbius fun tion fo r the pattern p
A very sho rt p rehisto ry Wilf (2002): Should b e done Wilf (2003): A mess. Don't tou h it. 16
SLIDE 18 Jason Smith (2014) A des ent in a p ermutation is a letter follo w ed b y a smaller
516792348 17
SLIDE 19 Jason Smith (2014) A des ent in a p ermutation is a letter follo w ed b y a smaller
516792348 18
SLIDE 20 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 19
SLIDE 21 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 20
SLIDE 22 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 21
SLIDE 23 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 The tail
an adja en y is all but its rst letter. 22
SLIDE 24 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 The tail
an adja en y is all but its rst letter. 23
SLIDE 25 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 The tail
an adja en y is all but its rst letter. An
a pattern in a p ermutation π is no rmal if the
all the tails
. 24
SLIDE 26 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 516792348 The tail
an adja en y is all but its rst letter. An
a pattern in a p ermutation π is no rmal if the
all the tails
. The no rmal
3412 in 516792348: 5734 and 6734 25
SLIDE 27 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 516792348 The tail
an adja en y is all but its rst letter. An
a pattern in a p ermutation π is no rmal if the
all the tails
. The no rmal
3412 in 516792348: 5734 and 6734 26
SLIDE 28 Jason Smith (2014) An adja en y in a p ermutation is a maximal in reasing se- quen e
letters adja ent in p
and value: 516792348 516792348 The tail
an adja en y is all but its rst letter. An
a pattern in a p ermutation π is no rmal if the
all the tails
. Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . 27
SLIDE 29 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . 28
SLIDE 30 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . 29
SLIDE 31 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. 30
SLIDE 32 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. That is, the
is shellable. 31
SLIDE 33 P
b d e f Order
− →
b d e f 32
SLIDE 34 P
b d e f Order
− →
b d e f 33
SLIDE 35 P
b d e f Order
− →
b d e f
− →
SLIDE 36 P
→
b d e f Order
− →
1
1
1
1
1
1 35
SLIDE 37 P
→
b d e f Order
− →
1
1
1
1
1
1 Contra tible A sphere 36
SLIDE 38 P
→
b d e f Order
− →
1
1
1
1
1
1 Contra tible A sphere The Mbius fun tion equals the redu ed Euler ha ra teristi 37
SLIDE 39
- Shellable
- mplex
- Nonshellable
- mplex
38
SLIDE 40
- Shellable
- mplex
- Nonshellable
- mplex
- 39
SLIDE 41
- Shellable
- mplex
- Nonshellable
- mplex
- X
40
SLIDE 42
- Shellable
- mplex
- Nonshellable
- mplex
- X
41
SLIDE 43
- Shellable
- mplex
- Nonshellable
- mplex
- µ(σ,τ )
equals redu ed Euler ha ra teristi
shellable
is homotopi ally a w edge
spheres.
redu ed Euler ha ra teristi is the numb er
spheres.
has nontrivial homology at most in the top dimension. 42
SLIDE 44 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. Pro
Bije t to sub w
and use Bj rner's results (1988). 43
SLIDE 45 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. Theo rem: Let π b e any p ermutation with a segment
three
numb ers in de reasing
in reasing
Then µ(1,π) = 0 . 44
SLIDE 46 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. Theo rem: Let π b e any p ermutation with a segment
three
numb ers in de reasing
in reasing
Then µ(1,π) = 0 .
µ(1, 71654823) = 0
45
SLIDE 47 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. Theo rem: Let π b e any p ermutation with a segment
three
numb ers in de reasing
in reasing
Then µ(1,π) = 0 . In fa t, the interval [1,π] is
46
SLIDE 48 Jason Smith (2014) Theo rem: If σ and τ have the same numb er
des ents, then
µ(σ, τ) = (−1)|τ|−|σ|N(σ, τ),
where N(σ, τ) is the numb er
no rmal
in τ . Therefo re,
|µ(σ, τ)| σ(τ),
where σ(τ) is the numb er
in τ . And, the interval [σ, τ] is shellable. There a re results/ onje tures analogous to the ab
fo r the la y ered and sepa rable p ermutations. 47
SLIDE 49 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations 48
SLIDE 50 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations 3 2 1 5 4 6 8 7 49
SLIDE 51 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations 3 2 1 5 4 6 8 7 50
SLIDE 52 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations 3 2 1 5 4 6 8 7 A la y ered p ermutation is a
de reasing se- quen es, ea h smaller than the next. 51
SLIDE 53 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations
2 1 5 4 6 8 7 A la y ered p ermutation is a
de reasing se- quen es, ea h smaller than the next. 52
SLIDE 54 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations
2 1 5 4 6 8 7 53
SLIDE 55 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations
2 1 5 4 6 8 7 (Any subsequen e
a la y ered p ermutation is la y ered) 54
SLIDE 56 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations
2 1 5 4 6 8 7 An ee tive fo rmula, but to
to t inside these ma rgins . . . 55
SLIDE 57 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations
2 1 5 4 6 8 7 An ee tive fo rmula, but to
to t inside these ma rgins . . . (Simila r to p ermutations with xed numb er
des ents) 56
SLIDE 58 Sagan-V atter (2005): Mbius fun tion fo r la y ered p ermutations
2 1 5 4 6 8 7 A sp e ial ase
the sepa rable p ermutations. 57
SLIDE 61 4 2 3 5 1 7 8 6
de omp
p ermutation is a dire t sum
42351786 = 42351 ⊕ 231
60
SLIDE 62 4 2 3 5 1 7 8 6
de omp
p ermutation is a dire t sum
42351786 = 42351 ⊕ 231
A sk ew-de omp
p ermutation is a sk ew sum
76841325 = 213 ⊖ 41325
61
SLIDE 63 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 62
SLIDE 64 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 63
SLIDE 65 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 64
SLIDE 66 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 65
SLIDE 67 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 66
SLIDE 68 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 67
SLIDE 69 4 2 3 5 1 7 8 6
p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 68
SLIDE 70 4 2 3 5 1 7 8 6
rable A p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. 69
SLIDE 71 4 2 3 5 1 7 8 6
rable A p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. De omp
b y sk ew/dire t sums into singletons 70
SLIDE 72 4 2 3 5 1 7 8 6
rable A p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. De omp
b y sk ew/dire t sums into singletons A p ermutation is sepa rable if and
if it avoids the pat- terns 2413 and 3142. 71
SLIDE 73 4 2 3 5 1 7 8 6
rable A p ermutation is sepa rable if it an b e generated from 1 b y dire t sums and sk ew sums. De omp
b y sk ew/dire t sums into singletons A p ermutation is sepa rable if and
if it avoids the pat- terns 2413 and 3142. 2 4 1 3
sepa rable 72
SLIDE 74 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . 73
SLIDE 75 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . 74
SLIDE 76 Nan Li −
→
L Bru e Sagan −
→
S 75
SLIDE 77 Nan Li −
→
L Bru e Sagan −
→
S E. Babson, A. Bj rner, L, V. W elk er, J. Sha reshian 76
SLIDE 78 Nan Li −
→
L Bru e Sagan −
→
S 77
SLIDE 79 Nan Li −
→
L Bru e Sagan −
→
S Lou Billera −
→
B 78
SLIDE 80 Nan Li −
→
L Bru e Sagan −
→
S Lou Billera −
→
B Mi helle W a hs −
→
MW 79
SLIDE 81 Nan Li −
→
L Bru e Sagan −
→
S Lou Billera −
→
B Mi helle W a hs −
→
MW P eter M Nama ra −
→
M N 80
SLIDE 82 Nan Li −
→
L Bru e Sagan −
→
S Lou Billera −
→
B Mi helle W a hs −
→
MW P eter M Nama ra −
→
M N Abb reviating y
last name to a single letter implies every- b
should rememb er y
name. 81
SLIDE 83 Nan Li −
→
L Bru e Sagan −
→
S Lou Billera −
→
B Mi helle W a hs −
→
MW P eter M Nama ra −
→
M N Abb reviating y
last name to a single letter implies every- b
should rememb er y
name. Let's put an end to this immo dest y! 82
SLIDE 84 Nan Li −
→
L Bru e Sagan −
→
S Lou Billera −
→
B Mi helle W a hs −
→
MW P eter M Nama ra −
→
M N Abb reviating y
last name to a single letter implies every- b
should rememb er y
name. Let's put an end to this immo dest y! (Unless y
name is Central Ship y a rd, in whi h ase y
ma y b e fo rgiven) 83
SLIDE 85 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . 84
SLIDE 86 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . (This
in p
time) 85
SLIDE 87 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . Co rolla ry: If σ and τ a re sepa rable then
|µ(σ,τ )| σ(τ )
where σ(τ ) is the numb er
in τ . 86
SLIDE 88 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . Co rolla ry: If σ and τ a re sepa rable then
|µ(σ,τ )| σ(τ )
where σ(τ ) is the numb er
in τ . (A generalization
a
T enner and Steingrms- son) 87
SLIDE 89 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . Co rolla ry: If σ and τ a re sepa rable then
|µ(σ,τ )| σ(τ )
where σ(τ ) is the numb er
in τ .
µ(135 . . . (2k
- 1) (2k) . . . 42, 135 . . . (2n -1) (2n) . . . 42) =
n+k−1
n−k
SLIDE 90 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . Co rolla ry: If σ and τ a re sepa rable then
|µ(σ,τ )| σ(τ )
where σ(τ ) is the numb er
in τ .
µ(1342, 13578642) =
8/2+4/2−1
8/2−4/2
5
2
SLIDE 91 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . Co rolla ry: If σ and τ a re sepa rable then
|µ(σ,τ )| σ(τ )
where σ(τ ) is the numb er
in τ . Co rolla ry: If τ is sepa rable, then µ( 1, τ) ∈ {0, 1, −1} . 90
SLIDE 92 Burstein, Jelnek, Jelnk
Steingrmsson: Theo rem: If σ and τ a re sepa rable p ermutations, then
µ(σ,τ ) =
(
1)parity(X) where the sum is
unpaired
in τ . Co rolla ry: If σ and τ a re sepa rable then
|µ(σ,τ )| σ(τ )
where σ(τ ) is the numb er
in τ . Co rolla ry: If τ is sepa rable, then µ( 1, τ) ∈ {0, 1, −1} . Neither
ry true in general 91
SLIDE 93 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter. 92
SLIDE 94 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter.
P
anti hain:
1 2 3 4 · · ·
1344 P 113414 1343 P 113414
93
SLIDE 95 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter.
P
anti hain:
1 2 3 4 · · ·
1344 P 113414 1343 P 113414
94
SLIDE 96 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter.
P
anti hain:
1 2 3 4 · · ·
1344 P 113414 1343 P 113414 P
hain: . . .
1343 P 113414
95
SLIDE 97 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter.
P
anti hain:
1 2 3 4 · · ·
1344 P 113414 1343 P 113414 P
hain: . . .
1343 P 113414
(3 <P 4 ) 96
SLIDE 98 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter.
P
anti hain:
1 2 3 4 · · ·
1344 P 113414 1343 P 113414 P
hain: . . .
1343 P 113414
Fixed-des La y ered 97
SLIDE 99 La y ered intervals and xed-des intervals a re isomo rphi to t w
in the Generalized sub w
(determined b y a p
)
Sagan and V atter.
P
anti hain:
1 2 3 4 · · ·
1344 P 113414 1343 P 113414 P
hain: . . .
1343 P 113414
Fixed-des La y ered Is there a family
intervals
p ermutations interp
b et w een these t w
(that a re shellable,
at least with a tra table Mbius fun tion)? 98
SLIDE 100 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else 99
SLIDE 101 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else Co rolla ry: If σ is inde omp
then µ(σ, τ) = 0 unless
τ = τ1 ⊕ τ2 ⊕ · · · ⊕ τk
τ = τ1 ⊕ τ2 ⊕ · · · ⊕ τk ⊕ 1
. 100
SLIDE 102 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else Co rolla ry: If σ is inde omp
then µ(σ, τ) = 0 unless
τ = τ1 ⊕ τ2 ⊕ · · · ⊕ τk
τ = τ1 ⊕ τ2 ⊕ · · · ⊕ τk ⊕ 1
. Co rolla ry: If σ = σ1 ⊕ σ2 and
τ = τ1 ⊕ τ2
a re nest,
τ1, τ2 > 1
and τ1 = τ2 , then µ(σ, τ) = µ(σ1, τ1) · µ(σ2, τ2) . 101
SLIDE 103 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else Co rolla ry: If σ is inde omp
then µ(σ, τ) = 0 unless
τ = τ1 ⊕ τ2 ⊕ · · · ⊕ τk
τ = τ1 ⊕ τ2 ⊕ · · · ⊕ τk ⊕ 1
. Co rolla ry: If σ = σ1 ⊕ σ2 and
τ = τ1 ⊕ τ2
a re nest,
τ1, τ2 > 1
and τ1 = τ2 , then µ(σ, τ) = µ(σ1, τ1) · µ(σ2, τ2) . (only sometimes this is b e ause [σ, τ] is a dire t p ro du t) 102
SLIDE 104 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else 103
SLIDE 105 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else This redu es the
the Mbius fun tion to inde omp
p ermutations. 104
SLIDE 106 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else This redu es the
the Mbius fun tion to inde omp
p ermutations. Unfo rtunately , almost all p ermutations a re inde omp
and w e have no idea ho w to deal with them in general . . . 105
SLIDE 107 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else This redu es the
the Mbius fun tion to inde omp
p ermutations. Unfo rtunately , almost all p ermutations a re inde omp
and w e have no idea ho w to deal with them in general . . . X 106
SLIDE 108 M Nama ra-Steingrmsson (refo rmulation
BJJS): Theo rem: Let τ = τ 1 ⊕ · · · ⊕ τ k b e nest de omp
Then
µ(σ,τ ) =
µ(σm,τ m) + ǫm
where
ǫm =
1,
if σm = ∅ and τ m−1 = τ m
0,
else This redu es the
the Mbius fun tion to inde omp
p ermutations. Unfo rtunately , almost all p ermutations a re inde omp
and w e have no idea ho w to deal with them in general . . . X 107
SLIDE 109 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three. 108
SLIDE 110 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three.
2136547 215436 3216547 2154367 2154376 1326547 21437658 14327658 21543768 21543876 215438769
A dis onne ted interval
rank 3 109
SLIDE 111 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three.
2136547 215436 3216547 2154367 2154376 1326547 21437658 14327658 21543768 21543876 215438769
A dis onne ted interval
rank 3 110
SLIDE 112 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three. 111
SLIDE 113 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three. Theo rem: Almost every interval has a dis onne ted subin- terval
rank at least three. 112
SLIDE 114 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three. Theo rem: Almost every interval has a dis onne ted subin- terval
rank at least three. F
ws from the Stanley-Wilf
The numb er
p ermutations avoiding any given pattern p gro ws
exp
. 113
SLIDE 115 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three. Theo rem: Almost every interval has a dis onne ted subin- terval
rank at least three. F
ws from the Ma r us-T a rdos theo rem: The numb er
p ermutations avoiding any given pattern p gro ws
exp
. 114
SLIDE 116 An
to shellabilit y
an interval is having a dis-
subinterval
rank at least three. Theo rem: Almost every interval has a dis onne ted subin- terval
rank at least three. F
ws from the Ma r us-T a rdos theo rem: The numb er
p ermutations avoiding any given pattern p gro ws
exp
. Thus, almost every interval [σ,τ ] (fo r τ la rge enough)
the subintervals [π,π ⊕π] and [π,π ⊖π] fo r some
π > 1
,
whi h is dis onne ted. 115
SLIDE 117 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3. 116
SLIDE 118 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3.
1 5 4 3 6 2 1 5 4 3 8 7 6 9 117
SLIDE 119 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3.
1 5 4 3 6 2 1 5 4 3 8 7 6 9 118
SLIDE 120 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3.
1 5 4 3 6 2 1 5 4 3 8 7 6 9 [215436, 215438769℄ is dis onne ted 119
SLIDE 121 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3.
1 5 4 3 6 2 1 5 4 3 8 7 6 9 [215436, 215438769℄ is dis onne ted 120
SLIDE 122 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3.
2136547 215436 3216547 2154367 2154376 1326547 21437658 14327658 21543768 21543876 215438769
[215436, 215438769℄ is dis onne ted 121
SLIDE 123 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3.
2136547 215436 3216547 2154367 2154376 1326547 21437658 14327658 21543768 21543876 215438769
[215436, 215438769℄ is dis onne ted 122
SLIDE 124 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3. 123
SLIDE 125 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3. Theo rem: An interval
la y ered p ermutations is shellable if and
if it has no dis onne ted subintervals
rank 3
mo re. 124
SLIDE 126 M Nama ra and Steingrmsson: Theo rem: An interval [σ, τ]
la y ered p ermutations is dis-
if and
if σ and τ dier b y a rep eated la y er
size at least 3. Theo rem: An interval
la y ered p ermutations is shellable if and
if it has no dis onne ted subintervals
rank 3
mo re. Conje ture: The same is true
sepa rable p ermutations. 125
SLIDE 127 The interval
[123, 3416725]
has no non-trivial dis onne ted subintervals, and alternating Mbius fun tion, but homology in dierent dimensions. Betti numb ers: 0, 1, 2. 126
SLIDE 128 Some questions:
p rop
intervals have µ = 0 ? 127
SLIDE 129 Some questions:
p rop
intervals have µ = 0 ? Almost all? 128
SLIDE 130 Some questions:
p rop
intervals have µ = 0 ? Almost all?
kinds
intervals exist in P 129
SLIDE 131 Some questions:
p rop
intervals have µ = 0 ? Almost all?
kinds
intervals exist in P ? T
130
SLIDE 132 Some questions:
p rop
intervals have µ = 0 ? Almost all?
kinds
intervals exist in P ? T
there to rsion in the homology
any intervals? 131
SLIDE 133 Some questions:
p rop
intervals have µ = 0 ? Almost all?
kinds
intervals exist in P ? T
there to rsion in the homology
any intervals?
the rank fun tion
every interval unimo dal? 132
SLIDE 134 Some questions:
p rop
intervals have µ = 0 ? Almost all?
kinds
intervals exist in P ? T
there to rsion in the homology
any intervals?
the rank fun tion
every interval unimo dal?
w do es max(|µ(1, π)|) gro w with the length
? 133
SLIDE 135 Thanks, Ri ha rd! (and y
all ¨
⌣ )
134
