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On some properties of permutation tableaux Alexander Burstein Iowa - - PowerPoint PPT Presentation

Oct 23, 2023 758 likes •2.4k views

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems On some properties of permutation tableaux Alexander Burstein Iowa State University burstein@math.iastate.edu 4th

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

On some properties of permutation tableaux

Alexander Burstein Iowa State University burstein@math.iastate.edu 4th International Conference on Permutation Patterns June 15, 2006

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

1

Permutation Tableaux Row and Column Labeling Tableaux and Permutations

2

Row and Column Decomposition

3

Statistics on Tableaux and Permutations Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

4

Essential 1s Distribution Bare Tableaux

5

Open Problems

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau

Alex Burstein On some properties of permutation tableaux

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SLIDE 4

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau is a filling of a k × (n − k) rectangle

Alex Burstein On some properties of permutation tableaux

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SLIDE 5

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau is a filling of a k × (n − k) rectangle with 0s, 1s and 2s

Alex Burstein On some properties of permutation tableaux

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SLIDE 6

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau is a filling of a k × (n − k) rectangle with 0s, 1s and 2s such that

Alex Burstein On some properties of permutation tableaux

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SLIDE 7

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau is a filling of a k × (n − k) rectangle with 0s, 1s and 2s such that (SE-2s) The 2s cut out a Ferrers board.

Alex Burstein On some properties of permutation tableaux

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SLIDE 8

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau is a filling of a k × (n − k) rectangle with 0s, 1s and 2s such that (SE-2s) The 2s cut out a Ferrers board. (1-hinge) A cell must contain a 1 if there is a 1 to its left in the same row and a 1 above it in the same column.

Alex Burstein On some properties of permutation tableaux

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SLIDE 9

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Definition of Permutation Tableau

A permutation tableau is a filling of a k × (n − k) rectangle with 0s, 1s and 2s such that (SE-2s) The 2s cut out a Ferrers board. (1-hinge) A cell must contain a 1 if there is a 1 to its left in the same row and a 1 above it in the same column. (column) Every column contains at least one 1.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10

Alex Burstein On some properties of permutation tableaux

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SLIDE 11

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Alex Burstein On some properties of permutation tableaux

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SLIDE 12

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 1 2 2 2 2 2

Alex Burstein On some properties of permutation tableaux

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SLIDE 13

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 2 2 2 1 1 2 2 2 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2

Alex Burstein On some properties of permutation tableaux

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SLIDE 14

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 2 2 2 1 1 2 2 2 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2

Alex Burstein On some properties of permutation tableaux

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SLIDE 15

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 2 0 0 0 0 1 1 1 1 1 2 0 0 1 0 1 1 1 1 1 2 0 0 0 0 0 1 1 2 2 2 0 0 0 0 0 1 1 2 2 2 1 1 1 0 1 1 2 2 2 2 0 1 1 1 1 2 2 2 2 2

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1

Alex Burstein On some properties of permutation tableaux

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SLIDE 17

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

n = 20, k = 10

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Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Boundary Path

Alex Burstein On some properties of permutation tableaux

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SLIDE 20

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Boundary Path

Extend the 01/2 boundary to a SW lattice path (n − k, 0) → (0, k) on n steps.

Alex Burstein On some properties of permutation tableaux

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SLIDE 21

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Boundary Path

Extend the 01/2 boundary to a SW lattice path (n − k, 0) → (0, k) on n steps. Label the path steps 1 through n from NE to SW end.

Alex Burstein On some properties of permutation tableaux

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SLIDE 22

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Boundary Path

Extend the 01/2 boundary to a SW lattice path (n − k, 0) → (0, k) on n steps. Label the path steps 1 through n from NE to SW end. Label the rows and columns with the label of the corresponding step.

Alex Burstein On some properties of permutation tableaux

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SLIDE 23

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Boundary Path

Extend the 01/2 boundary to a SW lattice path (n − k, 0) → (0, k) on n steps. Label the path steps 1 through n from NE to SW end. Label the rows and columns with the label of the corresponding step. Label each cell p with its row and column labels (lr(p), lc(p)).

Alex Burstein On some properties of permutation tableaux

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SLIDE 24

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Boundary Path

Extend the 01/2 boundary to a SW lattice path (n − k, 0) → (0, k) on n steps. Label the path steps 1 through n from NE to SW end. Label the rows and columns with the label of the corresponding step. Label each cell p with its row and column labels (lr(p), lc(p)). Note that lc(p) > lr(p) for every cell p in the tableau.

Alex Burstein On some properties of permutation tableaux

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SLIDE 25

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

Alex Burstein On some properties of permutation tableaux

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SLIDE 26

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4 Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Example

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T

Alex Burstein On some properties of permutation tableaux

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SLIDE 29

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths

Alex Burstein On some properties of permutation tableaux

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SLIDE 30

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths start at NW boundary

Alex Burstein On some properties of permutation tableaux

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SLIDE 31

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths start at NW boundary salient points at 1s

Alex Burstein On some properties of permutation tableaux

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SLIDE 32

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths start at NW boundary salient points at 1s

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths start at NW boundary salient points at 1s

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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SLIDE 34

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths start at NW boundary salient points at 1s

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

starting with a tableau T use SE paths start at NW boundary salient points at 1s

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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SLIDE 36

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

Alex Burstein On some properties of permutation tableaux

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SLIDE 37

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

  • 1

2 3 4 5 6 7 8 9 10 17 7 18 3 8 11 15 5 4 12 11 12 13 14 15 16 17 18 19 20 13 2 20 6 19 10 16 9 1 14

  • Alex Burstein

On some properties of permutation tableaux

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SLIDE 38

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

  • 1

2 3 4 5 6 7 8 9 10 17 7 18 3 8 11 15 5 4 12 11 12 13 14 15 16 17 18 19 20 13 2 20 6 19 10 16 9 1 14

  • Φ is a bijection.

Alex Burstein On some properties of permutation tableaux

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SLIDE 39

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

  • 1

2 3 4 5 6 7 8 9 10 17 7 18 3 8 11 15 5 4 12 11 12 13 14 15 16 17 18 19 20 13 2 20 6 19 10 16 9 1 14

  • Φ is a bijection.

How to find Φ−1?

Alex Burstein On some properties of permutation tableaux

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SLIDE 40

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

  • 1

2 3 4 5 6 7 8 9 10 17 7 18 3 8 11 15 5 4 12 11 12 13 14 15 16 17 18 19 20 13 2 20 6 19 10 16 9 1 14

  • Φ is a bijection.

How to find Φ−1? Steingr´ ımsson, Williams 2005: Reconstruction of T from π by columns from right to left.

Alex Burstein On some properties of permutation tableaux

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SLIDE 41

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

  • 1

2 3 4 5 6 7 8 9 10 17 7 18 3 8 11 15 5 4 12 11 12 13 14 15 16 17 18 19 20 13 2 20 6 19 10 16 9 1 14

  • Φ is a bijection.

How to find Φ−1? Steingr´ ımsson, Williams 2005: Reconstruction of T from π by columns from right to left. Reconstruction by rows from bottom to top is similar.

Alex Burstein On some properties of permutation tableaux

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SLIDE 42

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row and Column Labeling Tableaux and Permutations

Φ : Tableaux → Permutations

π = Φ(T) =

  • 1

2 3 4 5 6 7 8 9 10 17 7 18 3 8 11 15 5 4 12 11 12 13 14 15 16 17 18 19 20 13 2 20 6 19 10 16 9 1 14

  • Φ is a bijection.

How to find Φ−1? Steingr´ ımsson, Williams 2005: Reconstruction of T from π by columns from right to left. Reconstruction by rows from bottom to top is similar. We will reconstruct T from π by columns from left to right (and by rows from top to bottom).

Alex Burstein On some properties of permutation tableaux

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SLIDE 43

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column.

Alex Burstein On some properties of permutation tableaux

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SLIDE 44

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point.

Alex Burstein On some properties of permutation tableaux

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SLIDE 45

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i1, . . . , ik be the row labels of 1s in the leftmost column.

Alex Burstein On some properties of permutation tableaux

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SLIDE 46

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i1, . . . , ik be the row labels of 1s in the leftmost column. Then π(n) < π(i1) < π(i2) < · · · < π(ik) = n ...

Alex Burstein On some properties of permutation tableaux

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SLIDE 47

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i1, . . . , ik be the row labels of 1s in the leftmost column. Then π(n) < π(i1) < π(i2) < · · · < π(ik) = n ... ...and for j ∈ (ir, ir+1), π(j) < π(ir).

Alex Burstein On some properties of permutation tableaux

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SLIDE 48

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i1, . . . , ik be the row labels of 1s in the leftmost column. Then π(n) < π(i1) < π(i2) < · · · < π(ik) = n ... ...and for j ∈ (ir, ir+1), π(j) < π(ir). Hence, π(i1), π(i2), . . . , π(ik) = n are the LR-minima of the subsequence of π on letters greater than π(n).

Alex Burstein On some properties of permutation tableaux

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SLIDE 49

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Leftmost column

We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i1, . . . , ik be the row labels of 1s in the leftmost column. Then π(n) < π(i1) < π(i2) < · · · < π(ik) = n ... ...and for j ∈ (ir, ir+1), π(j) < π(ir). Hence, π(i1), π(i2), . . . , π(ik) = n are the LR-minima of the subsequence of π on letters greater than π(n). If T ′ is T without the leftmost column, then Φ(T) = Φ(T ′)(i1 i2 . . . ik n).

Alex Burstein On some properties of permutation tableaux

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SLIDE 50

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

Alex Burstein On some properties of permutation tableaux

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SLIDE 51

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j,

Alex Burstein On some properties of permutation tableaux

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SLIDE 52

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j, ci contains b,

Alex Burstein On some properties of permutation tableaux

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SLIDE 53

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j, ci contains b, cj contains a < c,

Alex Burstein On some properties of permutation tableaux

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SLIDE 54

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j, ci contains b, cj contains a < c, a < b < c,

Alex Burstein On some properties of permutation tableaux

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SLIDE 55

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j, ci contains b, cj contains a < c, a < b < c,

then cj also contains b.

Alex Burstein On some properties of permutation tableaux

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SLIDE 56

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Column Decomposition

The column decomposition of a permutation π is the (unique) representation of π as a product cn−kcn−k−1 . . . c1 of increasing cycles ci (1 ≤ i ≤ n − k) such that maximal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j, ci contains b, cj contains a < c, a < b < c,

then cj also contains b. This condition is equivalent to the 1-hinge rule for tableaux.

Alex Burstein On some properties of permutation tableaux

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SLIDE 57

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = Alex Burstein On some properties of permutation tableaux

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SLIDE 59

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (3 4)(5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

π = (3 4)(5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19)(1 3 13 20) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Column Decomposition

π = (3 4)(5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19)(1 3 13 20) 1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Row Decomposition

The row decomposition of a permutation π is the (unique) representation of π as a product ckck−1 . . . c1 of decreasing cycles ci (1 ≤ i ≤ k) such that minimal elements of ci’s are distinct from one another and from other elements in ci’s, and if

i < j, ci contains b, cj contains c > a, c > b > a,

then cj also contains b. This condition is also equivalent to the 1-hinge rule for tableaux.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Example of Row Decomposition

π = (19 18 17 16 14 15)(20 19 18 16 14 13)(14 12 11)(14 12 10)(18 16 14 12 9 8 7)(16 14 12 9 8 6)(12 9 8 5)(20 19 18 14 12 9 4 3) (14 12 2)(20 19 1) 1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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Alignments and Crossings

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Alignments and Crossings

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Intersections: ee, nn, en (e < n)

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Alignments and Crossings

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

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Intersections: ee, nn, en (e < n) Non-intersections: ne (n < e)

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Alignments and Crossings

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nn nn nn nn nn nn nn nn ee ee en en en nn nn nn nn nn nn ee ee en en en en ne ee ee en en ne ee ee nn ne ee ee ee en en ne ne ne ee ee ee en en ne ne ne en ne ne ne ne ee ne ne ne ne ne

Intersections: ee, nn, en (e < n) Non-intersections: ne (n < e)

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Statistics on Permutation Tableaux

Let π = Φ(T). Define

AEE(π) = |{(i, j) | j < i ≤ π(i) < π(j)}| = #ee(T) ANN(π) = |{(i, j) | π(j) < π(i) < i < j}| = #nn(T) AEN(π) = |{(i, j) | j ≤ π(j) < π(i) < i}| = #en(T) ANE(π) = |{(i, j) | π(i) < i < j ≤ π(j)}| = #ne(T) = #2s(T) CEE(π) = |{(i, j) | j < i ≤ π(j) < π(i)}| CNN(π) = |{(i, j) | π(i) < π(j) < i < j}|

It can be shown that CEE(π) + CNN(π) = #nontop 1s(T)

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Steingr´ ımsson and Williams define a map Ψ : Sn → Sn that takes descents (des) and ascents (ndes) to weak excedances (wex) and deficiencies (nwex).

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Steingr´ ımsson and Williams define a map Ψ : Sn → Sn that takes descents (des) and ascents (ndes) to weak excedances (wex) and deficiencies (nwex). They show that Ψ has the following properties. If π = Ψ(σ), then des σ = wex π − 1 (31-2)σ = AEE(π) + ANN(π) (21-3)σ + (3-21)σ − des σ 2

  • = AEN(π)

(2-31)σ = CEE(π) + CNN(π) (1-32)σ + (32-1)σ − des σ 2

  • = ANE(π)

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

This implies: AEN + ANN and CEE + CNN are equidistributed, AEN and ANE are equidistributed.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

This implies: AEN + ANN and CEE + CNN are equidistributed, AEN and ANE are equidistributed. To see the latter note that the map i ◦ r ◦ c (inverse of reversal of complement, or reflection across the antidiagonal of the permutation diagram) preserves wex, AEE, ANN, CEE, CNN, and exchanges AEN and ANE.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

This implies: AEN + ANN and CEE + CNN are equidistributed, AEN and ANE are equidistributed. To see the latter note that the map i ◦ r ◦ c (inverse of reversal of complement, or reflection across the antidiagonal of the permutation diagram) preserves wex, AEE, ANN, CEE, CNN, and exchanges AEN and ANE. Question: Describe the equivalent (under Φ) of irc on tableaux directly.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Catalan tableaux

Recall that |Sn(2-31)| = |Sn(2-3-1)| = Cn, the nth Catalan number.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Catalan tableaux

Recall that |Sn(2-31)| = |Sn(2-3-1)| = Cn, the nth Catalan number. Hence, (SW, 2005) Cn is the number of tableaux with no nontop 1s (i.e. with a single 1 per column).

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Catalan tableaux

Theorem If a tableau T has a single 1 per column, and π = Φ(T), then (the underlying sets of) the cycles of π form a noncrossing partition.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Catalan tableaux

Theorem If a tableau T has a single 1 per column, and π = Φ(T), then (the underlying sets of) the cycles of π form a noncrossing partition. Proof. T has a single 1 per column, so no element of π may occur in more than one cycle of its row decomposition. Suppose π contains two cycles ci and cj and elements a > b > c > d such that a, c are in ci and b, d are in cj.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Catalan tableaux

Theorem If a tableau T has a single 1 per column, and π = Φ(T), then (the underlying sets of) the cycles of π form a noncrossing partition. Proof. T has a single 1 per column, so no element of π may occur in more than one cycle of its row decomposition. Suppose π contains two cycles ci and cj and elements a > b > c > d such that a, c are in ci and b, d are in cj. If i < j, then cj contains c. If i > j, then ci contains b. Neither is possible.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Catalan tableaux

Theorem If a tableau T has a single 1 per column, and π = Φ(T), then (the underlying sets of) the cycles of π form a noncrossing partition. Proof. T has a single 1 per column, so no element of π may occur in more than one cycle of its row decomposition. Suppose π contains two cycles ci and cj and elements a > b > c > d such that a, c are in ci and b, d are in cj. If i < j, then cj contains c. If i > j, then ci contains b. Neither is possible. Thus, the ci’s form a noncrossing partition.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Note that AEE(T) = ANN(T) = 0 for monotone T.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Note that AEE(T) = ANN(T) = 0 for monotone T. Let π = Φ(T) and let σ = Ψ−1(π).

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Note that AEE(T) = ANN(T) = 0 for monotone T. Let π = Φ(T) and let σ = Ψ−1(π). Then σ avoids 31-2 (i.e. 3-1-2).

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Note that AEE(T) = ANN(T) = 0 for monotone T. Let π = Φ(T) and let σ = Ψ−1(π). Then σ avoids 31-2 (i.e. 3-1-2). Note that the subsequences of weak excedance values of π and deficiency values of π are increasing.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Note that AEE(T) = ANN(T) = 0 for monotone T. Let π = Φ(T) and let σ = Ψ−1(π). Then σ avoids 31-2 (i.e. 3-1-2). Note that the subsequences of weak excedance values of π and deficiency values of π are increasing. Hence, π avoids 3-2-1.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Alignments and Crossings vs. Pattern Statistics Catalan tableaux Monotone tableaux

Monotone tableaux

A monotone tableau is one that has no 0 below or to the right

  • f any 1.

Note that AEE(T) = ANN(T) = 0 for monotone T. Let π = Φ(T) and let σ = Ψ−1(π). Then σ avoids 31-2 (i.e. 3-1-2). Note that the subsequences of weak excedance values of π and deficiency values of π are increasing. Hence, π avoids 3-2-1. Theorem Sn(31-2) Ψ → Sn(3-2-1) Φ ← monotone tableaux of semiperimeter n.

Alex Burstein On some properties of permutation tableaux

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Definition

Essential 1s –

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Definition

Essential 1s – leftmost 1s in their rows

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Definition

Essential 1s – leftmost 1s in their rows or topmost 1s in their columns.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Definition

Essential 1s – leftmost 1s in their rows or topmost 1s in their columns. Doubly essential 1s –

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Definition

Essential 1s – leftmost 1s in their rows or topmost 1s in their columns. Doubly essential 1s – leftmost 1s in their rows

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Definition

Essential 1s – leftmost 1s in their rows or topmost 1s in their columns. Doubly essential 1s – leftmost 1s in their rows and topmost 1s in their columns.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Example

  • •
  • •
  • • •
  • • •
  • • •
  • • • • •
  • • • • •
  • •
  • •
  • • •
  • •
  • • • •

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Example

⊚ ◦ ⊚ ◦

  • • ◦
  • • ◦
  • • ◦

⊚ • • • •

  • • • • •
  • •
  • •
  • • •
  • •
  • • ◦ •

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Statistics

Define statistics:

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Statistics

Define statistics: ess(T) = number of essential 1s in a tableau T.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Statistics

Define statistics: ess(T) = number of essential 1s in a tableau T. dess(T) = number of doubly essential 1s in a tableau T.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Statistics

Define statistics: ess(T) = number of essential 1s in a tableau T. dess(T) = number of doubly essential 1s in a tableau T. Note that n − ess(T) = dess(T) + zerorows(T).

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Distribution

Conjecture (Steingr´ ımsson, Williams, 2005) (n − ess, rows) has the same distribution on tableaux as (cycles, wex) on permutations.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Distribution

Theorem (B., Eriksen, 2006) (dess + zerorows, rows) has the same distribution on tableaux as (cycles, wex) on permutations.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Distribution

Theorem (B., Eriksen, 2006) (dess + zerorows, rows) has the same distribution on tableaux as (cycles, wex) on permutations. Each 0-row adds 1 to each of dess + zerorows, rows, cycles, wex, so we only need to consider tableaux with no 0-rows vs. derangements.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Bare tableaux

Define a new type of tableau called a bare tableau.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Bare tableaux

Define a new type of tableau called a bare tableau. Almost all rules are the same as for permutation tableaux.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Bare tableaux

Define a new type of tableau called a bare tableau. Almost all rules are the same as for permutation tableaux except (1-hinge) A cell must contain a 1 if there is a 1 to its left in the same row and a 1 above it in the same column.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Bare tableaux

Define a new type of tableau called a bare tableau. Almost all rules are the same as for permutation tableaux except (0-hinge) A cell must contain a 0 if there is a 1 to its left in the same row and a 1 above it in the same column.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Example

  • •
  • •
  • • •
  • • •
  • • •
  • • • • •
  • • • • •
  • •
  • •
  • • •
  • •
  • • • •

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Example

⊚ ◦ ⊚ ◦

  • • ◦
  • • ◦
  • • ◦

⊚ • • • •

  • • • • •
  • •
  • •
  • • •
  • •
  • • ◦ •

Alex Burstein On some properties of permutation tableaux

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SLIDE 117

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Example

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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SLIDE 118

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Properties of the filling map

The filling map (0-hinge → 1-hinge) is a bijection:

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Properties of the filling map

The filling map (0-hinge → 1-hinge) is a bijection:

essential 1s are uniquely determined by the tableaux;

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Properties of the filling map

The filling map (0-hinge → 1-hinge) is a bijection:

essential 1s are uniquely determined by the tableaux; essential 1s uniquely determine the nonessential 1s.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Properties of the filling map

The filling map (0-hinge → 1-hinge) is a bijection:

essential 1s are uniquely determined by the tableaux; essential 1s uniquely determine the nonessential 1s.

The filling map preserves:

Alex Burstein On some properties of permutation tableaux

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SLIDE 122

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Properties of the filling map

The filling map (0-hinge → 1-hinge) is a bijection:

essential 1s are uniquely determined by the tableaux; essential 1s uniquely determine the nonessential 1s.

The filling map preserves:

positions of weak excedances (wexb) and non-weak-excedances (nwext);

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Properties of the filling map

The filling map (0-hinge → 1-hinge) is a bijection:

essential 1s are uniquely determined by the tableaux; essential 1s uniquely determine the nonessential 1s.

The filling map preserves:

positions of weak excedances (wexb) and non-weak-excedances (nwext); positions of 1 and n.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • Alex Burstein

On some properties of permutation tableaux

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SLIDE 131

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • ⊚ ◦
  • ◦

1 3 7 13 15 20 19 18 17 9 4 Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • ⊚ ◦
  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • ⊚ ◦
  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

6 16

⊚

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • ⊚ ◦
  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

6 16

⊚

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Decomposition

1 2 3 5 6 7 10 11 13 15 20 19 18 17 16 14 12 9 8 4

⊚ ◦ ⊚ ◦

  • ⊚
  • ⊚ ◦
  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

6 16

⊚

(1 7 18 9 4 3 13 20 17 15 19) (2 10 11 14 8 5 12) (6 16) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

“Barely” labeled binary trees

⊚ ◦

  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

6 16

⊚

(1 7 18 9 4 3 13 20 17 15 19) (2 10 11 14 8 5 12) (6 16) Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

“Barely” labeled binary trees

⊚ ◦

  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

6 16

⊚

(1 7 18 9 4 3 13 20 17 15 19) (2 10 11 14 8 5 12) (6 16) 1 20 3 13 18 7 9 4 19 15 17 Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

“Barely” labeled binary trees

⊚ ◦

  • ◦

1 3 7 13 15 20 19 18 17 9 4 2 5 10 11 14 12 8

⊚

  • ◦

6 16

⊚

(1 7 18 9 4 3 13 20 17 15 19) (2 10 11 14 8 5 12) (6 16) 1 20 3 13 18 7 9 4 19 15 17 2 14 10 11 12 5 8 6 16 Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Bare labeling properties

If a vertex is a left child, its label is the smallest in its subtree.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Bare labeling properties

If a vertex is a left child, its label is the smallest in its subtree. If a vertex is a right child, its label is the largest in its subtree.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Tree traversal (barely labeled tree → cycle)

Start from the smallest label (at the root) along the left edge, if possible. If there is no left child, this is the first return to the root (see last rule).

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Tree traversal (barely labeled tree → cycle)

Start from the smallest label (at the root) along the left edge, if possible. If there is no left child, this is the first return to the root (see last rule). At each step, start at the previous vertex and

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Tree traversal (barely labeled tree → cycle)

Start from the smallest label (at the root) along the left edge, if possible. If there is no left child, this is the first return to the root (see last rule). At each step, start at the previous vertex and

try to move away from the root alternating unused left and right edges as long as possible.

Alex Burstein On some properties of permutation tableaux

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SLIDE 144

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Tree traversal (barely labeled tree → cycle)

Start from the smallest label (at the root) along the left edge, if possible. If there is no left child, this is the first return to the root (see last rule). At each step, start at the previous vertex and

try to move away from the root alternating unused left and right edges as long as possible.

  • therwise (if there are no such edges) move towards the root

along the same-side edges as long as possible.

Alex Burstein On some properties of permutation tableaux

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SLIDE 145

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Tree traversal (barely labeled tree → cycle)

Start from the smallest label (at the root) along the left edge, if possible. If there is no left child, this is the first return to the root (see last rule). At each step, start at the previous vertex and

try to move away from the root alternating unused left and right edges as long as possible.

  • therwise (if there are no such edges) move towards the root

along the same-side edges as long as possible.

The label of the end vertex of this path is the next term in the cycle.

Alex Burstein On some properties of permutation tableaux

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SLIDE 146

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Tree traversal (barely labeled tree → cycle)

Start from the smallest label (at the root) along the left edge, if possible. If there is no left child, this is the first return to the root (see last rule). At each step, start at the previous vertex and

try to move away from the root alternating unused left and right edges as long as possible.

  • therwise (if there are no such edges) move towards the root

along the same-side edges as long as possible.

The label of the end vertex of this path is the next term in the cycle. At the first return to the root, the next term is the largest label at the

  • root. At the second return to the root (and when the root has no right

child), the cycle is complete.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row).

Alex Burstein On some properties of permutation tableaux

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SLIDE 148

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

Alex Burstein On some properties of permutation tableaux

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SLIDE 149

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17 Alex Burstein On some properties of permutation tableaux

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SLIDE 150

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17

(1 7 18 9 4 3 13 20 17 15 19)

Alex Burstein On some properties of permutation tableaux

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SLIDE 151

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17

(1 7 18 9 4 3 13 20 17 15 19)

Alex Burstein On some properties of permutation tableaux

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SLIDE 152

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17

(1 7 18 9 4 3 13 20 17 15 19) Successive minima (maxima) in the direction of inverse cycle.

Alex Burstein On some properties of permutation tableaux

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SLIDE 153

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17

(1 7 18 9 4 3 13 20 17 15 19) Successive minima (maxima) in the direction of inverse cycle in reverse order.

Alex Burstein On some properties of permutation tableaux

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SLIDE 154

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17

(1 7 18 9 4 3 13 20 17 15 19) Successive minima (maxima) in the direction of inverse cycle in reverse order.

Alex Burstein On some properties of permutation tableaux

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SLIDE 155

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Distribution Bare Tableaux

Cycle → barely labeled tree

Need only find 1s in the leftmost column (or in the top row). ...i.e. successive left (or right) children starting from the root.

1 20 3 13 18 7 9 4 19 15 17

(1 7 18 9 4 3 13 20 17 15 19) Successive minima (maxima) in the direction of inverse cycle in reverse order.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Take your pick.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Take your pick. Plus avoidance: even more.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Take your pick. Plus avoidance: even more. Steingr´ ımsson, Williams, B., Eriksen, Reifegerste, Viennot: some answers.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Take your pick. Plus avoidance: even more. Steingr´ ımsson, Williams, B., Eriksen, Reifegerste, Viennot: some answers. Let B be “Φ on bare tableaux”. How is B−1(π) related to Φ−1(π)?

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Take your pick. Plus avoidance: even more. Steingr´ ımsson, Williams, B., Eriksen, Reifegerste, Viennot: some answers. Let B be “Φ on bare tableaux”. How is B−1(π) related to Φ−1(π)? Let aij be the number of derangements π such that B−1(π) has i essential 1s, and Φ−1(π) has j essential 1s. Let A = [aij]. What can be said about A?

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

Open Problems

Take your pick. Plus avoidance: even more. Steingr´ ımsson, Williams, B., Eriksen, Reifegerste, Viennot: some answers. Let B be “Φ on bare tableaux”. How is B−1(π) related to Φ−1(π)? Let aij be the number of derangements π such that B−1(π) has i essential 1s, and Φ−1(π) has j essential 1s. Let A = [aij]. What can be said about A? Etc.

Alex Burstein On some properties of permutation tableaux

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Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems

  • E. Babson, E. Steingr´

ımsson, Generalized permutation patterns and a classification of the Mahonian statistics, S´

  • em. Lothar. Combin., B44b (2000), 18 pp.
  • M. B´
  • na. Combinatorics of Permutations, Chapman & Hall/CRC Press, 2004.
  • A. Burstein, S. Elizalde, T. Mansour, Restricted Dumont permutations, Dyck paths, and noncrossing

partitions, to appear in Proceedings of the 18th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC’06), June 2006, San Diego, CA, USA. Online at http://garsia.math.yorku.ca/fpsac06/papers/78_ps_or_pdf.pdf.

  • A. Claesson: Generalized pattern avoidance, Europ. J. Combin. 22 (2001), 961–971.
  • S. Corteel, Crossing and alignments of permutations, preprint, arXiv:math.CO/0601469, 13 pp.
  • S. Corteel, Crossings and alignments of permutations, preprint, arXiv:math.CO/0505031, 11 pp.
  • A. Postnikov, Webs in totally positive Grassmann cells, in preparation.
  • A. Postnikov, personal communication.
  • E. Steingr´

ımsson, L.K. Williams, Permutation tableaux and permutation patterns, preprint, arXiv:math.CO/0507149, 21 pp.

  • L. Williams, Enumeration of totally positive Grassmann cells, Advances in Math., 190 (2005), 319-342.

Alex Burstein On some properties of permutation tableaux