Staircases , dominoes and leaves : Bounds on gr(Av(1324)) David - - PowerPoint PPT Presentation

▶
staircases dominoes and leaves bounds on gr av 1324
SMART_READER_LITE
LIVE PREVIEW

Staircases , dominoes and leaves : Bounds on gr(Av(1324)) David - - PowerPoint PPT Presentation

Jan 20, 2023 32 likes •565 views

Staircases , dominoes and leaves : Bounds on gr(Av(1324)) David Bevan University of Strathclyde (Based on joint work with Robert Brignall , Andrew Elvey Price & Jay Pantone ) Permutation Patterns 2017 , Hsklinn Reykjavk, sland 29

slide-1
SLIDE 1

Staircases, dominoes and leaves: Bounds on gr(Av(1324))

David Bevan

University of Strathclyde

(Based on joint work with Robert Brignall, Andrew Elvey Price & Jay Pantone)

Permutation Patterns 2017, Háskólinn í Reykjavík, Ísland 29th June 2017

slide-2
SLIDE 2

Bounds on the growth rate of 1324-avoiders

Av(1324) is the only unenumerated class avoiding a pattern of length 4. gr(C) = lim

n→∞ |Cn|1/n

Lower Upper 2004: Bóna 288 2005: Bóna 9 2006: Albert et al. 9.47 2012: Claesson, Jelínek & Steingrímsson† 16 2014: Bóna 13.93 2015: Bóna 13.74 2015: B. 9.81 2015: Conway & Guttmann estimate gr(Av(1324)) ≈ 11.60 ± 0.01

†An upper bound of 13.002 follows from an unproven conjecture.

slide-3
SLIDE 3

Bounds on the growth rate of 1324-avoiders

Av(1324) is the only unenumerated class avoiding a pattern of length 4. gr(C) = lim

n→∞ |Cn|1/n

Lower Upper 2004: Bóna 288 2005: Bóna 9 2006: Albert et al. 9.47 2012: Claesson, Jelínek & Steingrímsson† 16 2014: Bóna 13.93 2015: Bóna 13.74 2015: B. 9.81 This work 10.27 13.5 2015: Conway & Guttmann estimate gr(Av(1324)) ≈ 11.60 ± 0.01

†An upper bound of 13.002 follows from an unproven conjecture.

slide-4
SLIDE 4

The staircase

The infinite decreasing

  • Av(213), Av(132)
  • staircase:

S =

Av(132) Av(213) Av(132) Av(213) Av(132) Av(213)

Proposition

Av(1324) ⊂ S

slide-5
SLIDE 5

Gridding a 1324-avoider

slide-6
SLIDE 6

Gridding a 1324-avoider

Av(213)

p1

  • p1 uppermost 1 in a 213
slide-7
SLIDE 7

Gridding a 1324-avoider

Av(213) Av(132)

p1 p2

  • p1 uppermost 1 in a 213
  • p2 leftmost 2 in a 132 consisting of points below p1 divider
slide-8
SLIDE 8

Gridding a 1324-avoider

Av(213) Av(132)

p1 p2

  • p2 leftmost 2 in a 132 consisting of points below p1 divider
  • No points above p1 and to the right of p2
slide-9
SLIDE 9

Gridding a 1324-avoider

Av(213) Av(132) Av(213)

p1 p2 p3

  • p2 leftmost 2 in a 132 consisting of points below p1 divider
  • p3 uppermost 1 in a 213 consisting of points to right of p2 divider
slide-10
SLIDE 10

Gridding a 1324-avoider

Av(213) Av(132) Av(213)

p1 p2 p3

  • p3 uppermost 1 in a 213 consisting of points to right of p2 divider
  • No points to the left of p2 and below p3
slide-11
SLIDE 11

Gridding a 1324-avoider

Av(213) Av(132) Av(213) Av(132)

p1 p2 p3 p4

  • p3 uppermost 1 in a 213 consisting of points to right of p2 divider
  • p4 leftmost 2 in a 132 consisting of points below p3 divider
slide-12
SLIDE 12

Gridding a 1324-avoider

Av(213) Av(132) Av(213) Av(132)

p1 p2 p3 p4

  • p4 leftmost 2 in a 132 consisting of points below p3 divider
  • No points above p3 and to the right of p4
slide-13
SLIDE 13

Gridding a 1324-avoider

Av(213) Av(132) Av(213) Av(132) Av(213)

  • Terminates after no more than n/2 steps.
slide-14
SLIDE 14

Gridding a 1324-avoider

Av(213) Av(132) Av(213) Av(132) Av(213)

  • Terminates after no more than n/2 steps.
  • Gridding is greedy: each cell contains as many points as possible.
slide-15
SLIDE 15

Gridding a 1324-avoider

Av(213) Av(132) Av(213) Av(132) Av(213)

  • Hasse graph of Av(213) is skew-decomposable forest of up-trees.
  • Hasse graph of Av(132) is skew-decomposable forest of down-trees.
slide-16
SLIDE 16

The greedy gridding of a large 1324-avoider

Data provided by Einar Steingrímsson.

slide-17
SLIDE 17

Dominoes

A domino is a gridded permutation in

Av(132) Av(213)

that avoids 1324. = =

slide-18
SLIDE 18

Dominoes

A domino is a gridded permutation in

Av(132) Av(213)

that avoids 1324. = = Important: / ∈ D (D = the set of dominoes)

slide-19
SLIDE 19

Dominoes

A domino is a gridded permutation in

Av(132) Av(213)

that avoids 1324. = = Important: / ∈ D (D = the set of dominoes)

slide-20
SLIDE 20

Dominoes

Theorem

The number of n-point dominoes is 2(3n + 3)! (n + 2)!(2n + 3)!. gr(D) = 27/4.

slide-21
SLIDE 21

Dominoes

Theorem

The number of n-point dominoes is 2(3n + 3)! (n + 2)!(2n + 3)!. gr(D) = 27/4.

Proof.

Bijection between dominoes and certain arch configurations.

  • Functional equation solved using iterated discriminants.
slide-22
SLIDE 22

Dominoes

Theorem

The number of n-point dominoes is 2(3n + 3)! (n + 2)!(2n + 3)!. gr(D) = 27/4.

Definition

Jay (v. tr.) To confirm a conjectural enumeration by asking Jay Pantone. Example: “I doubt that this can be Jayed.” Jayable (adj.) Example: “Perhaps this sequence is Jayable.”

slide-23
SLIDE 23

Dominoes

Theorem

The number of n-point dominoes is 2(3n + 3)! (n + 2)!(2n + 3)!. gr(D) = 27/4.

A familiar sequence

Dominoes are equinumerous to

  • West-2-stack-sortable permutations
  • Rooted nonseparable planar maps

Open problem

Find a bijection between dominoes and another combinatorial structure.

slide-24
SLIDE 24

Mapping a 1324-avoider to a domino

  • Greedy grid the permutation.
  • Interpret the gridded permutation as a sequence of dominoes.
  • Use Φ to construct a large domino, splitting the small dominoes.
  • Φ is not injective; it discards vertical interleaving information.

Φ :

Av(132) Av(213) Av(132) Av(213) Av(132) Av(213)

→

Av(132) Av(213) Av(132) Av(213) Av(132) Av(213)

slide-25
SLIDE 25

Upper bound

Ψ : Avn(1324) → (•, •)n × Dn →

slide-26
SLIDE 26

Upper bound

Ψ : Avn(1324) → (•, •)n × Dn →

slide-27
SLIDE 27

Upper bound

Ψ : Avn(1324) → (•, •)n × Dn →

  • The vertical interleaving can be recovered from the •• sequence.
slide-28
SLIDE 28

Upper bound

Ψ : Avn(1324) → (•, •)n × Dn →

  • The vertical interleaving can be recovered from the •• sequence.
  • Ψ is an injection.

gr(Av(1324)) 2 × 27/4 = 13.5

slide-29
SLIDE 29

Lower bound (1)

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213) Av(132) Av(213) Av(213)

  • Decompose staircase into dominoes and connecting cells.
slide-30
SLIDE 30

Lower bound (1)

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213)

  • To avoid 1324, position blue/red points

between yellow/green skew indecomposable components.

slide-31
SLIDE 31

Lower bound (1)

Av(132) p points Av(132) p points Av(132) p points Av(213) p points Av(213) p points Av(213) p points

4p 7 pts p 2 comps 4p 7 pts p 2 comps 4p 7 pts p 2 comps

  • Optimal values yield

gr(Av(1324)) 81/8 = 10.125

slide-32
SLIDE 32

Leaves of a domino

Leaves

  • left-to-right minima of lower cell
  • right-to-left maxima of upper cell
slide-33
SLIDE 33

Leaves of a domino

Leaves

  • left-to-right minima of lower cell
  • right-to-left maxima of upper cell
  • leaves of trees in Hasse graphs
slide-34
SLIDE 34

Leaves of a domino

Leaves

  • left-to-right minima of lower cell
  • right-to-left maxima of upper cell

Theorem

The expected number of leaves in an n-point domino is 5n/9.

slide-35
SLIDE 35

Better control of the interleaving

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213)

slide-36
SLIDE 36

Better control of the interleaving

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213)

slide-37
SLIDE 37

Better control of the interleaving

  • If every non-leaf occurs between yellow/green components,

leaves can be arbitrarily interleaved without creating a 1324.

slide-38
SLIDE 38

Better control of the interleaving

  • w1 = 2

w2 = 0 w3 = 1 w4 = 2 w5 = 0 w6 = 1

  • The non-leaves split a cell into a sequence of strips.
  • wi: width of ith strip (number of leaves in strip)
slide-39
SLIDE 39

Better control of the interleaving

  • w1 = 2

w2 = 0 w3 = 1 w4 = 2 w5 = 0 w6 = 1

  • wi: width of ith strip (number of leaves in strip)

Theorem

The expected number of empty strips in an n-point domino is 5n/27.

slide-40
SLIDE 40

Better control of the interleaving

  • Which strip widths minimise the number of ways of interleaving?
slide-41
SLIDE 41

Better control of the interleaving

  • Which strip widths minimise the number of ways of interleaving?

Lemma

The worst case for interleaving is when the leaves are distributed as evenly as possibly between the strips.

slide-42
SLIDE 42

Better control of the interleaving

  • Which strip widths minimise the number of ways of interleaving

◮ if at least 5/

9 of the points are leaves

◮ and at least 5/

12 = (5/ 27)/(4/ 9) of the strips are empty?

Lemma

The worst case for interleaving is when the leaves are distributed as evenly as possibly between the strips.

slide-43
SLIDE 43

Better control of the interleaving

  • Which strip widths minimise the number of ways of interleaving

◮ if at least 5/

9 of the points are leaves

◮ and at least 5/

12 = (5/ 27)/(4/ 9) of the strips are empty?

  • 5/

12 of the strips are empty

  • no strips have 1 leaf
  • 1/

2 of the strips have 2 leaves

  • 1/

12 of the strips have 3 leaves

Lemma

The worst case for interleaving is when the leaves are distributed as evenly as possibly between the strips.

slide-44
SLIDE 44

Lower bound (2)

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213)

slide-45
SLIDE 45

Lower bound (2)

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213)

  • gr(Av(1324)) 10.27101292824530 . . .
slide-46
SLIDE 46

Lower bound (2)

Av(132) Av(132) Av(132) Av(213) Av(213) Av(213)

  • gr(Av(1324)) 10.27101292824530 . . .
  • This value is a root of a polynomial of degree 104,

whose smallest coefficient has 86 digits.

slide-47
SLIDE 47

Questions for the future

  • Conjecture of Claesson, Jelínek & Steingrímsson
slide-48
SLIDE 48

Questions for the future

  • Conjecture of Claesson, Jelínek & Steingrímsson
  • Bijection between dominoes and something else
slide-49
SLIDE 49

Questions for the future

  • Conjecture of Claesson, Jelínek & Steingrímsson
  • Bijection between dominoes and something else
  • Improvement on the (crude) upper bound
slide-50
SLIDE 50

Questions for the future

  • Conjecture of Claesson, Jelínek & Steingrímsson
  • Bijection between dominoes and something else
  • Improvement on the (crude) upper bound
  • Expected proportion of k-leaf strips
slide-51
SLIDE 51

Questions for the future

  • Conjecture of Claesson, Jelínek & Steingrímsson
  • Bijection between dominoes and something else
  • Improvement on the (crude) upper bound
  • Expected proportion of k-leaf strips
  • Trominoes

Av(213) Av(213) Av(132) Av(213) Av(132) Av(132)

slide-52
SLIDE 52

Questions for the future

  • Conjecture of Claesson, Jelínek & Steingrímsson
  • Bijection between dominoes and something else
  • Improvement on the (crude) upper bound
  • Expected proportion of k-leaf strips
  • Trominoes
  • “Turning the corner” seems to require new ideas

Av(213) Av(213) Av(132) Av(213) Av(132) Av(132)

slide-53
SLIDE 53

Thanks!