A simple explicit bijection between ( n , 2 ) Gog and Magog - - PowerPoint PPT Presentation

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A simple explicit bijection between ( n , 2 ) Gog and Magog - - PowerPoint PPT Presentation

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6 6 6 6 6 6 6 6 6 Introduction The bijection A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids Jrmie B ETTINELLI March 7, 2016 Jrmie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog

slide-1
SLIDE 1 6 6 6 6 6

Introduction

6 6 6 6

The bijection

A simple explicit bijection between (n, 2) Gog and Magog trapezoids

Jérémie BETTINELLI

March 7, 2016

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-2
SLIDE 2 5 6 6 6 6

Introduction

6 6 6 6

The bijection

What are Gog and Magog?

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-3
SLIDE 3 5 6 6 6 6

Introduction

6 6 6 6

The bijection

What are Gog and Magog?

In the mathematical world, these are combinatorial objects known to be in bijection with other fundamental objects.

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

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SLIDE 4 6 5 6 6 6

Introduction

6 6 6 6

The bijection

Alternating sign matrices

Definition

An alternating sign matrix of size n is an n × n matrix with entries in {−1, 0, 1} such that, on each fixed row or column, the nonzero entries start and end by 1 and alternate between 1 and -1.     1 1 −1 1 1 1    

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-5
SLIDE 5 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1
  • 1

Alternating sign matrices

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-6
SLIDE 6 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1
  • 1

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-7
SLIDE 7 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1
  • 1

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-8
SLIDE 8 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1

  • 1
  • 1
  • 1

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-9
SLIDE 9 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1

  • 1
  • 1
  • 1

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-10
SLIDE 10 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1

  • 1
  • 1
  • 1

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-11
SLIDE 11 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1
  • 1

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-12
SLIDE 12 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-13
SLIDE 13 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

6-vertex model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-14
SLIDE 14 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

loop model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-15
SLIDE 15 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

even coordinates

  • dd coordinates

loop model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-16
SLIDE 16 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

loop model

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-17
SLIDE 17 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1
  • 1

Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-18
SLIDE 18 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1
  • 1

Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-19
SLIDE 19 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1

  • 1

Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-20
SLIDE 20 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1 1

  • 1

Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-21
SLIDE 21 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1 1 1 1 Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-22
SLIDE 22 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

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SLIDE 23 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 4 4 1 1 1 1 2 4 5 5 Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-24
SLIDE 24 6 6 5 6 6

Introduction

6 6 6 6

The bijection

Gog

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 4 4 1 1 1 1 2 4 5 5 Gog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

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SLIDE 25 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Totally symmetric self-complementary plane partitions

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-26
SLIDE 26 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Totally symmetric self-complementary plane partitions

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-27
SLIDE 27 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Totally symmetric self-complementary plane partitions

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-28
SLIDE 28 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Non intersecting lattice paths

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-29
SLIDE 29 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Non intersecting lattice paths

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-30
SLIDE 30 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Non intersecting lattice paths

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-31
SLIDE 31 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

Non intersecting lattice paths

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-32
SLIDE 32 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

− + − − + − + + − − + − − − + − − Magog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-33
SLIDE 33 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

− + − − + − + + − − + − − − + − − Magog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-34
SLIDE 34 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

− + − − + − + + − − + − − − + − − Magog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-35
SLIDE 35 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

− + − − + − + + − − + − − − + − − Magog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-36
SLIDE 36 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

− − + − + + − − + − − − + − − 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 4 Magog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-37
SLIDE 37 6 6 6 5 6

Introduction

6 6 6 6

The bijection

Magog

1 2 3 4 5 6 − − + − + + − − + − − − + − − 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 4 Magog

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

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SLIDE 38 6 6 6 6 5

Introduction

6 6 6 6

The bijection

Trapezoids

1 2 3 3 4 4 5 5 6 6 7 7 8 8 2 2 4 5 6 7 8 1 1 2 4 4 5 7 7 1 2 2 4 4 6 7 7 1 1 2 4 4 5 7 (8, 2) Gog trapezoid (8, 2) Magog trapezoid

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-39
SLIDE 39 6 6 6 6 5

Introduction

6 6 6 6

The bijection

Trapezoids

1 2 3 3 4 4 5 5 6 6 7 7 8 8 2 2 4 5 6 7 8 1 1 2 4 4 5 7 7 1 2 2 4 4 6 7 7 1 1 2 4 4 5 7 (8, 2) Gog trapezoid (8, 2) Magog trapezoid

Theorem (Zeilberger ’96)

(n, k) Magog trapezoids and (n, k) Gog trapezoids are equinumerous.

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-40
SLIDE 40 6 6 6 6 5

Introduction

6 6 6 6

The bijection

Trapezoids

1 2 3 3 4 4 5 5 6 6 7 7 8 8 2 2 4 5 6 7 8 1 1 2 4 4 5 7 7 1 2 2 4 4 6 7 7 1 1 2 4 4 5 7 (8, 2) Gog trapezoid (8, 2) Magog trapezoid

Theorem (Zeilberger ’96)

(n, k) Magog trapezoids and (n, k) Gog trapezoids are equinumerous.

6 k = 1

Bijection by Krattenthaller matching refined statistics

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-41
SLIDE 41 6 6 6 6 5

Introduction

6 6 6 6

The bijection

Trapezoids

1 2 3 3 4 4 5 5 6 6 7 7 8 8 2 2 4 5 6 7 8 1 1 2 4 4 5 7 7 1 2 2 4 4 6 7 7 1 1 2 4 4 5 7 (8, 2) Gog trapezoid (8, 2) Magog trapezoid

Theorem (Zeilberger ’96)

(n, k) Magog trapezoids and (n, k) Gog trapezoids are equinumerous.

6 k = 1

Bijection by Krattenthaller matching refined statistics

6 k = 2

Bijection by Biane & Cheballah ’12 This talk

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

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SLIDE 42 6 6 6 6 6

Introduction

5 6 6 6

The bijection

From Magog to Gog

→

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-43
SLIDE 43 6 6 6 6 6

Introduction

5 6 6 6

The bijection

From Magog to Gog

→ 1 2 2 4 4 6 7 7 1 1 2 4 4 6 7

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

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SLIDE 44 6 6 6 6 6

Introduction

5 6 6 6

The bijection

From Magog to Gog

→ 1 2 2 4 4 6 7 7 1 1 2 4 4 6 7 leftmost a b such that b > a + 1

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-45
SLIDE 45 6 6 6 6 6

Introduction

5 6 6 6

The bijection

From Magog to Gog

→ 1 2 2 4 4 6 7 7 1 1 2 4 4 6 7

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-46
SLIDE 46 6 6 6 6 6

Introduction

5 6 6 6

The bijection

From Magog to Gog

→

+1 −2

→ 1 2 2 4 4 6 7 7 1 1 2 4 4 6 7 2 3 4 4 6 7 7 1 1 2 2 2 2 4 5

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-47
SLIDE 47 6 6 6 6 6

Introduction

6 5 6 6

The bijection

From Gog to Magog

←

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-48
SLIDE 48 6 6 6 6 6

Introduction

6 5 6 6

The bijection

From Gog to Magog

← 2 3 4 4 6 7 7 1 1 2 2 2 2 4 5

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-49
SLIDE 49 6 6 6 6 6

Introduction

6 5 6 6

The bijection

From Gog to Magog

← 2 3 4 4 6 7 7 1 1 2 2 2 2 4 5 rightmost a b such that a ≤ b + 1

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-50
SLIDE 50 6 6 6 6 6

Introduction

6 5 6 6

The bijection

From Gog to Magog

← 2 3 4 4 6 7 7 1 1 2 2 2 2 4 5

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-51
SLIDE 51 6 6 6 6 6

Introduction

6 5 6 6

The bijection

From Gog to Magog

←

−1 +2

← 1 2 2 4 4 6 7 7 1 1 2 4 4 6 7 2 3 4 4 6 7 7 1 1 2 2 2 2 4 5

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-52
SLIDE 52 6 6 6 6 6

Introduction

6 6 5 6

The bijection

Degenerate case

↔ no a b with b > a + 1 ↔ a b with a ≤ b + 1

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-53
SLIDE 53 6 6 6 6 6

Introduction

6 6 5 6

The bijection

Degenerate case

↔

+1

< ↔ 1 2 2 4 4 6 6 8 1 1 2 3 4 4 5 2 3 3 5 5 7 8 1 1 2 3 4 4 5 6 first possibility

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-54
SLIDE 54 6 6 6 6 6

Introduction

6 6 5 6

The bijection

Degenerate case

↔

+1 +1

= ↔ 1 2 2 4 4 6 6 6 1 1 2 3 4 4 5 2 3 3 5 5 7 7 1 1 2 3 4 4 5 7 second possibility

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-55
SLIDE 55 6 6 6 6 6

Introduction

6 6 6 5

The bijection

Extension to (ℓ, n, 2) trapezoids

4 5 6 6 7 7 8 8 9 9 10 10 11 11 4 5 5 7 9 9 10 2 2 4 5 5 7 9 9 2 4 5 5 6 6 9 10 2 3 3 3 6 6 8 (3, 8, 2) Gog trapezoid (3, 8, 2) Magog trapezoid The previous bijection can be trivially extended to (ℓ, n, 2) trapezoids.

Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016

slide-56
SLIDE 56 6 6 6 6 6

Introduction

6 6 6 6

The bijection Jérémie BETTINELLI A simple explicit bijection between (n, 2) Gog and Magog trapezoids March 7, 2016